Syllabus Vector Differentiation - Velocity and Acceleration - Gradient ...
Syllabus Vector Differentiation - Velocity and Acceleration - Gradient ...
Syllabus Vector Differentiation - Velocity and Acceleration - Gradient ...
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Engineering Mathematics - II - <strong>Vector</strong> Calculus - 2007<br />
<strong>Syllabus</strong><br />
<strong>Vector</strong> <strong>Differentiation</strong><br />
- <strong>Velocity</strong> <strong>and</strong> <strong>Acceleration</strong><br />
- <strong>Gradient</strong><br />
- Divergence<br />
- Curl<br />
- Laplacian<br />
- Solenoidal <strong>and</strong> Irrotational <strong>Vector</strong>s<br />
Recap<br />
Scalar product<br />
<br />
: a .<br />
<br />
b<br />
<br />
a1 b1<br />
a2<br />
b2<br />
a3<br />
b3<br />
<br />
|<br />
a | | b | cos<br />
,<br />
<br />
is<br />
the<br />
angle<br />
between<br />
<br />
a<br />
<strong>and</strong><br />
<br />
b<br />
Pr ojection :<br />
projection<br />
of<br />
<br />
a<br />
on<br />
<br />
b<br />
is<br />
<br />
a . b.<br />
Work done : If a force<br />
<br />
F displaces are provide from A to B then<br />
the work done is<br />
<br />
F . AB<br />
<strong>Vector</strong><br />
product :<br />
<br />
a<br />
<br />
<br />
b<br />
<br />
<br />
i<br />
a1<br />
b1<br />
<br />
j<br />
a2<br />
b2<br />
<br />
k<br />
a3<br />
b3<br />
<br />
| a | | b | sin<br />
<br />
<br />
n<br />
where<br />
<br />
n<br />
is<br />
unit<br />
vector<br />
perpendicular<br />
to<br />
both<br />
<br />
a<br />
<strong>and</strong><br />
<br />
b<br />
Moment: If the vecto r moment of a force<br />
<br />
F about a po int A<br />
through a point<br />
P<br />
is<br />
<br />
M<br />
<br />
<br />
AP<br />
<br />
<br />
F<br />
Scalar<br />
triple<br />
<br />
product : [ a ,<br />
<br />
b ,<br />
<br />
c ]<br />
<br />
<br />
a .( b<br />
<br />
<br />
c )<br />
a1<br />
b1<br />
c1<br />
a2<br />
c2<br />
c3<br />
a3<br />
b3<br />
c3<br />
Page 1 of 72
Engineering Mathematics - II - <strong>Vector</strong> Calculus - 2007<br />
Equation of line :<br />
Equations of a line pas sin g through (x1, y1, z1)<br />
<strong>and</strong><br />
direction<br />
ratios<br />
a, b, c are<br />
x x1 y y1<br />
z z<br />
<br />
1<br />
a b c<br />
Equation of a plane : is of the form<br />
ax + by + cz + d = 0 where a,b,c are the DR's of the normal.<br />
Introduction<br />
In this chapter the basic concepts of Differential Calculus of scalar functions are<br />
extended to vector functions.<br />
Here we study vector differentiation, gradient, divergence, curl, solenoidal <strong>and</strong><br />
irrational fields.<br />
Also we study about vector integration <strong>and</strong> verification of the integral theorems.<br />
<strong>Vector</strong> Function of a Scalar Variable<br />
<br />
Let f f1<br />
i f2<br />
j f3<br />
k where f1, f2<br />
<strong>and</strong> f3<br />
are<br />
functions<br />
of<br />
a<br />
var iable<br />
t<br />
is called a vector function.<br />
If f1, f2 <strong>and</strong> f3 are differentiable then we define<br />
<br />
d f f(t t) f(t)<br />
df <br />
1 df <br />
2 df <br />
lim i j <br />
3<br />
k<br />
dt t0<br />
t dt dt dt<br />
Similarly higher order derivatives can also be defined.<br />
Note:<br />
<br />
d f <br />
0<br />
dt<br />
iff<br />
<br />
f<br />
is<br />
a<br />
cons tan t<br />
vector .<br />
Page 2 of 72
Engineering Mathematics - II - <strong>Vector</strong> Calculus - 2007<br />
Rules of differentiation<br />
<br />
1. If g(t) is a scalar function of t <strong>and</strong> f (t) is a vector function of<br />
<br />
d d f d <br />
scalar t then ( f) f<br />
dt dt dt<br />
2. If<br />
<br />
<strong>and</strong><br />
<br />
are<br />
scalar<br />
cons tan ts,<br />
<br />
f<br />
<strong>and</strong><br />
<br />
g<br />
are vector<br />
functions of<br />
t<br />
then<br />
d<br />
dt<br />
<br />
( f<br />
<br />
<br />
g)<br />
<br />
<br />
d f<br />
<br />
dt<br />
<br />
<br />
d g<br />
<br />
dt<br />
3.<br />
d<br />
dt<br />
Proof:<br />
4.<br />
d<br />
dt<br />
Proof:<br />
<br />
( f . g)<br />
<br />
d g<br />
f .<br />
dt<br />
<br />
d f <br />
. g<br />
dt<br />
<br />
Let f f1<br />
i f2<br />
j f3<br />
k<br />
<br />
f . g<br />
<br />
g g1<br />
i g2<br />
j g3<br />
k<br />
f1 g1<br />
f2<br />
g2<br />
f3<br />
g3<br />
d dg1<br />
df1<br />
dg2<br />
df2<br />
dg3<br />
df3<br />
( f . g) f1 g1<br />
f2<br />
g2<br />
f3<br />
g3<br />
dt<br />
dt dt dt dt dt dt<br />
dg<br />
<br />
<br />
<br />
1 dg<br />
<br />
2 dg<br />
<br />
3 df<br />
<br />
1 df<br />
<br />
2 df<br />
f<br />
<br />
3<br />
1 f2<br />
f3<br />
g1<br />
g2<br />
g3<br />
<br />
dt dt dt dt dt dt <br />
<br />
d g<br />
f . <br />
dt<br />
<br />
( f <br />
df<br />
.<br />
dt<br />
<br />
g<br />
<br />
d g d f<br />
g) f <br />
dt dt<br />
<br />
Let f f1<br />
i f2<br />
j f3<br />
k<br />
<br />
g g1<br />
i g2<br />
j g3<br />
k<br />
<br />
g<br />
<br />
f g<br />
<br />
<br />
i<br />
f1<br />
g1<br />
<br />
j<br />
f2<br />
g2<br />
<br />
k<br />
f3<br />
g3<br />
Page 3 of 72
Engineering Mathematics - II - <strong>Vector</strong> Calculus - 2007<br />
<br />
f g i (f2<br />
g3<br />
g2<br />
f3)<br />
d<br />
dt<br />
<br />
<br />
( f g)<br />
<br />
d<br />
dt<br />
<br />
i (f2<br />
g3<br />
g2<br />
f3)<br />
<br />
<br />
<br />
d f<br />
i<br />
2<br />
<br />
dt<br />
g3<br />
f2<br />
dg3<br />
dt<br />
<br />
dg2<br />
dt<br />
f3<br />
g2<br />
df3<br />
<br />
<br />
dt <br />
<br />
<br />
d f<br />
i<br />
2<br />
<br />
dt<br />
dg<br />
g <br />
2<br />
3<br />
dt<br />
<br />
f3<br />
<br />
<br />
<br />
<br />
i<br />
<br />
f2<br />
<br />
dg3<br />
dt<br />
g2<br />
df3<br />
<br />
<br />
dt <br />
<br />
d f<br />
<br />
dt<br />
<br />
g<br />
<br />
<br />
f<br />
<br />
<br />
d g<br />
dt<br />
5.<br />
d<br />
dt<br />
<br />
<br />
<br />
<br />
f , g, h <br />
<br />
<br />
<br />
<br />
<br />
d<br />
f<br />
,<br />
dt<br />
<br />
<br />
<br />
<br />
<br />
g, h <br />
<br />
<br />
<br />
<br />
<br />
<br />
f<br />
<br />
<br />
,<br />
<br />
d g <br />
d h <br />
,h f , g ,<br />
dt<br />
dt<br />
<br />
<br />
<br />
<br />
<br />
Proof:<br />
d<br />
dt<br />
<br />
f ,<br />
<br />
<br />
g,<br />
<br />
h <br />
<br />
<br />
d<br />
dt<br />
<br />
<br />
<br />
f<br />
<br />
<br />
<br />
<br />
g .<br />
<br />
<br />
<br />
<br />
<br />
h <br />
<br />
<br />
d<br />
dt<br />
<br />
f<br />
<br />
<br />
<br />
<br />
g .<br />
<br />
<br />
<br />
h<br />
<br />
<br />
f<br />
<br />
<br />
<br />
<br />
g .<br />
<br />
<br />
<br />
d h<br />
dt<br />
<br />
d f<br />
<br />
dt<br />
<br />
<br />
<br />
g <br />
<br />
<br />
.<br />
<br />
h<br />
<br />
<br />
<br />
f<br />
<br />
<br />
<br />
<br />
d g <br />
.<br />
dt <br />
<br />
<br />
h<br />
<br />
<br />
<br />
f g .<br />
<br />
<br />
<br />
d h<br />
dt<br />
<br />
d<br />
f<br />
,<br />
dt<br />
<br />
<br />
<br />
<br />
<br />
g , h <br />
<br />
<br />
<br />
<br />
<br />
d g <br />
<br />
f , , h<br />
dt<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
f ,<br />
<br />
<br />
<br />
g,<br />
<br />
d h <br />
dt<br />
<br />
<br />
<br />
<br />
<br />
d f<br />
6. If | f | cons tan t then f . 0<br />
dt<br />
Proof:<br />
<br />
Let | f | <br />
C<br />
<br />
| f | a<br />
2<br />
where a is constant<br />
<br />
<br />
f .<br />
<br />
f<br />
<br />
cons tan t<br />
<br />
d<br />
dt<br />
<br />
f .<br />
<br />
<br />
<br />
f <br />
<br />
<br />
<br />
0<br />
Page 4 of 72
Engineering Mathematics - II - <strong>Vector</strong> Calculus - 2007<br />
<br />
d f d f <br />
f . . f 0<br />
dt dt<br />
<br />
d f<br />
f .<br />
dt<br />
0<br />
7. If<br />
<br />
f<br />
has<br />
a<br />
fixed<br />
direction<br />
then<br />
<br />
f<br />
<br />
d f <br />
0<br />
dt<br />
Proof:<br />
Let<br />
<br />
f<br />
<br />
<br />
f e<br />
Now<br />
<br />
f <br />
<br />
d f<br />
dt<br />
<br />
f e <br />
d<br />
dt<br />
<br />
(f e)<br />
<br />
df d e <br />
f e e 0<br />
dt dt<br />
f<br />
df<br />
dt<br />
<br />
(e e)<br />
df <br />
f (0)<br />
dt<br />
<br />
0<br />
8.If<br />
<br />
f<br />
<br />
<br />
d f<br />
dt<br />
<br />
<br />
0<br />
then<br />
<br />
f<br />
has<br />
a<br />
fixed<br />
direction.<br />
Equation of a Space Curve<br />
Let P (x,y,z) be any point in space then its position vector is<br />
<br />
OP<br />
<br />
<br />
r<br />
<br />
<br />
x i<br />
<br />
y j<br />
<br />
<br />
z k<br />
If x = x(t), y = y(t) <strong>and</strong> z = z(t) are functions of a scalar variable<br />
t then<br />
<br />
r r (t)<br />
describes<br />
a<br />
curve<br />
in<br />
space<br />
called<br />
the<br />
space<br />
curve.<br />
Page 5 of 72
Engineering Mathematics - II - <strong>Vector</strong> Calculus - 2007<br />
Note 1:<br />
<br />
d r<br />
dt<br />
represents<br />
the<br />
velocity<br />
<br />
v<br />
(rate<br />
of<br />
change<br />
of<br />
position<br />
with<br />
which<br />
the<br />
ter min al po int<br />
of<br />
<br />
r<br />
describes the<br />
curve )<br />
Note 2:<br />
<br />
d r<br />
dt<br />
is<br />
along<br />
the<br />
direction<br />
of<br />
the<br />
tan gent<br />
to<br />
the<br />
space<br />
curve<br />
at<br />
P (x, y, z)<br />
Note 3:<br />
The<br />
unit<br />
tan gent<br />
vector<br />
to<br />
the<br />
space<br />
curve<br />
is<br />
denoted by<br />
<br />
t<br />
<strong>and</strong><br />
is given by<br />
<br />
t<br />
<br />
<br />
d r<br />
dt<br />
<br />
d r<br />
| |<br />
dt<br />
Note 4:<br />
<br />
d v<br />
dt<br />
<br />
<br />
d<br />
2<br />
r<br />
<br />
is called the acceleration a<br />
dt<br />
2<br />
of<br />
the<br />
curve<br />
at time<br />
t.<br />
Unit Normal to a Space Curve<br />
Let A be a fixed point on the curve <strong>and</strong> s be the length of the arc AP where<br />
P(x,y,z) is a point on the space curve.<br />
Page 6 of 72
Engineering Mathematics - II - <strong>Vector</strong> Calculus - 2007<br />
Let<br />
<br />
t<br />
be<br />
the<br />
unit<br />
tan gent<br />
vector<br />
at<br />
P.<br />
Consider<br />
d.w.r.t s<br />
<br />
t . t<br />
1<br />
<br />
t<br />
.<br />
<br />
d t<br />
ds<br />
<br />
<br />
d t <br />
. t<br />
ds<br />
<br />
0<br />
<br />
<br />
d t<br />
2 t . <br />
ds<br />
0<br />
<br />
<br />
t<br />
<br />
d t<br />
.<br />
ds<br />
<br />
0<br />
<br />
d t<br />
ds<br />
is<br />
a normal<br />
to<br />
the<br />
space<br />
curve<br />
<strong>and</strong><br />
<br />
n<br />
<br />
<br />
d t<br />
ds<br />
<br />
d t<br />
| |<br />
ds<br />
is<br />
called<br />
the<br />
unit<br />
normal to<br />
the<br />
curve.<br />
Scalar <strong>and</strong> <strong>Vector</strong> Fields<br />
Scalar Valued Function<br />
Let P(x,y,z) which is a scalar is called a scalar point function.<br />
Note: A scalar point function is also called a scalar fixed . = (x,y,z)<br />
Ex.: (x,y,z) = x 2 + y 2 + z 2<br />
<strong>Vector</strong> Valued Function<br />
Page 7 of 72
Engineering Mathematics - II - <strong>Vector</strong> Calculus - 2007<br />
Let P(x,y,z) be a point. A vector point function is a vector whose<br />
components are real valued functions of x,y,z. A vector point function is called a<br />
vector<br />
field<br />
<br />
F .<br />
Ex:<br />
<br />
<br />
f f (x, y, z) f1<br />
i f2<br />
j f3<br />
k<br />
<br />
F xyz i x<br />
2<br />
y j yz k<br />
Differential<br />
operator<br />
:<br />
<br />
denotes<br />
the<br />
operation<br />
<br />
i<br />
x<br />
<br />
<br />
y<br />
<br />
j<br />
<br />
<br />
k<br />
z<br />
<br />
<br />
<br />
i<br />
x<br />
<br />
does<br />
not<br />
represent<br />
a<br />
vector<br />
if<br />
only<br />
defines<br />
the<br />
differential<br />
operator<br />
<strong>Gradient</strong> of a Scalar Field<br />
<br />
Let(x, y, z) be a scalar field, then i k i<br />
x z x<br />
is called the gradient of denoted by .<br />
The<br />
following<br />
are<br />
some<br />
results<br />
obtained<br />
by<br />
the<br />
definition<br />
of<br />
<br />
.<br />
1. ( )<br />
,<br />
is a scalar cons tan t.<br />
2. ( )<br />
,<br />
<strong>and</strong> are scalar fields.<br />
3. ( )<br />
,<br />
<strong>and</strong> are scalar cons tan ts<br />
4.() <br />
<br />
5.<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
2<br />
<br />
6. d dx dy dz<br />
x y z<br />
<br />
i<br />
<br />
<br />
<br />
x<br />
<br />
<br />
j<br />
<br />
y<br />
<br />
<br />
k<br />
<br />
<br />
z <br />
<br />
<br />
i<br />
dx<br />
<br />
<br />
<br />
<br />
j dy<br />
<br />
<br />
k dz<br />
<br />
<br />
d<br />
<br />
<br />
<br />
. d r<br />
Page 8 of 72
Engineering Mathematics - II - <strong>Vector</strong> Calculus - 2007<br />
Geometrical Interpretation<br />
Q(x, y, z)<br />
c represents a surface in space <strong>and</strong><br />
to the surface at any point.<br />
<br />
represents a normal<br />
Note 1: Since = c on the surface.<br />
<br />
d . d r<br />
<br />
0<br />
which<br />
shows<br />
that<br />
<br />
is<br />
perpendicular<br />
to<br />
the<br />
tan gent<br />
plane<br />
at<br />
any<br />
po int .<br />
Note<br />
2 : Unit<br />
vector<br />
along<br />
<br />
is<br />
denoted by<br />
<br />
n<br />
<br />
<br />
is<br />
| |<br />
called<br />
the<br />
unit<br />
normal<br />
to the surface (x,y,z) = c.<br />
Directional Derivative<br />
Let<br />
<br />
a<br />
be<br />
a<br />
vector<br />
inclined at<br />
an<br />
angle<br />
<br />
with<br />
<br />
then<br />
<br />
.<br />
<br />
a is<br />
called<br />
the<br />
directional<br />
derivative<br />
along<br />
<br />
a .<br />
Note : Maximum value<br />
of directional<br />
derivative<br />
is<br />
<br />
<br />
n | |<br />
Page 9 of 72
Engineering Mathematics - II - <strong>Vector</strong> Calculus - 2007<br />
Divergence of a <strong>Vector</strong> Field<br />
<br />
<br />
Let f f1<br />
i f2<br />
j f3<br />
k be a vector field then divergence of f<br />
denoted by<br />
div<br />
<br />
f<br />
or<br />
<br />
.<br />
f<br />
is<br />
defined<br />
as<br />
f<br />
f<br />
1 f2<br />
f <br />
3 i<br />
div. f . f <br />
x y z x<br />
Physical Interpretation<br />
If<br />
<br />
v the<br />
velocity<br />
of<br />
a moving<br />
fluid<br />
at<br />
a po int<br />
P<br />
at<br />
time<br />
t,<br />
then<br />
<br />
div v<br />
represents<br />
the rate at which the fluid moves out of a unit volume enclosing the point P.<br />
Solenoidal <strong>Vector</strong><br />
A<br />
vector<br />
field<br />
<br />
f<br />
is<br />
said<br />
to<br />
be<br />
solenoidal<br />
if<br />
<br />
div. f<br />
<br />
0.<br />
Properties<br />
<br />
1. div ( f g) div f div g , where f <strong>and</strong> g are vector fields.<br />
<br />
2. div ( f ) div f where is a scalar cons tan t.<br />
3.<br />
<br />
div ( f g)<br />
cons tan ts.<br />
<br />
<br />
div f <br />
<br />
div g<br />
where<br />
<br />
<strong>and</strong><br />
<br />
are<br />
scalar<br />
4.<br />
If<br />
<br />
<br />
is a scalar field <strong>and</strong> f is a vector field then div ( f ) div f .<br />
Proof:<br />
<br />
Let f f1<br />
i f2<br />
j f3<br />
k<br />
<br />
then f f1<br />
i f2<br />
j f3<br />
k<br />
<br />
div ( f ) ( f1 ) ( f2<br />
) ( f3)<br />
x y z<br />
<br />
f1<br />
x<br />
<br />
<br />
f1<br />
x<br />
<br />
f<br />
<br />
2<br />
y<br />
<br />
f2<br />
<br />
y<br />
<br />
f<br />
<br />
3<br />
y<br />
<br />
f3<br />
<br />
y<br />
<br />
<br />
<br />
<br />
<br />
f1<br />
<br />
<br />
x <br />
<br />
<br />
<br />
i<br />
x<br />
<br />
. f1<br />
i<br />
<br />
<br />
div f<br />
<br />
<br />
.<br />
f<br />
Page 10 of 72
Engineering Mathematics - II - <strong>Vector</strong> Calculus - 2007<br />
Laplacian of a scalar field <br />
Let<br />
then<br />
= (x,y,z) be a scalar field<br />
<br />
is a vector field<br />
<strong>and</strong><br />
div<br />
( )<br />
.<br />
<br />
<br />
<br />
x<br />
<br />
2<br />
<br />
<br />
x<br />
2<br />
<br />
<br />
x <br />
<br />
<br />
<br />
y<br />
<br />
<br />
<br />
<br />
2<br />
<br />
<br />
y<br />
2<br />
z<br />
2<br />
<br />
<br />
<br />
<br />
y <br />
<br />
<br />
z<br />
<br />
<br />
z <br />
denoted by<br />
<br />
2 <br />
is<br />
called<br />
the<br />
Laplacian<br />
of<br />
a<br />
scalar<br />
field.<br />
Note 1:<br />
<br />
2<br />
<br />
<br />
2<br />
x<br />
2<br />
<br />
2<br />
<br />
y<br />
2<br />
<br />
2<br />
<br />
z<br />
2<br />
Note 2:<br />
A<br />
scalar<br />
field<br />
<br />
is<br />
called<br />
a harmonic<br />
function<br />
if<br />
<br />
2<br />
<br />
<br />
0<br />
Note 3:<br />
<br />
2<br />
() <br />
2<br />
, where <br />
2<br />
<strong>and</strong> are cons tan ts,<br />
<br />
<strong>and</strong><br />
<br />
are scalar fields.<br />
Curl of a <strong>Vector</strong> field<br />
<br />
Let f f1<br />
i f2<br />
j f3<br />
k<br />
then<br />
curl<br />
of<br />
<br />
f<br />
denoted by<br />
curl<br />
<br />
f<br />
or<br />
<br />
<br />
<br />
f<br />
is<br />
defined<br />
as<br />
<br />
<br />
<br />
f<br />
<br />
<br />
i j k<br />
/ x / y<br />
/ z<br />
f1 f2<br />
f3<br />
<br />
<br />
f <br />
<br />
3 f<br />
<br />
2<br />
i<br />
y z <br />
Physical Interpretation<br />
<br />
If v is the velocity of a particle in rigid body rotating about a fixed<br />
axis<br />
then<br />
the<br />
angular<br />
velocity<br />
at<br />
any<br />
po int<br />
is<br />
given by<br />
1<br />
2<br />
<br />
curl v .<br />
Page 11 of 72
Engineering Mathematics - II - <strong>Vector</strong> Calculus - 2007<br />
Irrational <strong>Vector</strong><br />
<br />
<br />
f is said to irrational if curl f 0<br />
Properties<br />
(i)<br />
If<br />
<br />
f<br />
<strong>and</strong><br />
<br />
g<br />
are<br />
vector<br />
fields then<br />
<br />
curl ( f<br />
<br />
<br />
g)<br />
<br />
curl<br />
<br />
f<br />
<br />
curl<br />
<br />
g<br />
<br />
<br />
(ii) Curl ( f ) Curl f , where is a scalar cons tan t.<br />
(iii)<br />
If<br />
<br />
is<br />
a<br />
scalar<br />
field<br />
<strong>and</strong><br />
<br />
f<br />
is<br />
vector<br />
field<br />
then<br />
<br />
curl ( f )<br />
<br />
<br />
curl<br />
<br />
f<br />
<br />
<br />
<br />
<br />
f<br />
Proof:<br />
<br />
<br />
Curl ( f ) ( f3)<br />
( f2<br />
) i<br />
<br />
y z <br />
<br />
<br />
<br />
f<br />
<br />
<br />
3<br />
y<br />
<br />
f <br />
2<br />
i<br />
z <br />
<br />
<br />
f3<br />
y<br />
<br />
<br />
f2<br />
i <br />
z <br />
<br />
<br />
f f <br />
<br />
<br />
Curl ( f ) <br />
3<br />
<br />
2<br />
i f3<br />
f2<br />
i ...(1)<br />
y z y z <br />
Consider<br />
<br />
<br />
<br />
f<br />
<br />
<br />
i j k<br />
/ x / y / z<br />
f1 f2<br />
f3<br />
<br />
<br />
<br />
f3<br />
y<br />
<br />
<br />
f2<br />
<br />
z <br />
<br />
i<br />
<br />
<br />
Curl ( f )<br />
<br />
<br />
curl<br />
<br />
f<br />
<br />
<br />
f<br />
(iv)<br />
<br />
Curl ( )<br />
0<br />
Proof:<br />
Curl ( )<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
i<br />
y z z y <br />
<br />
0<br />
<br />
<br />
(v). If f is a vector field then div (Curl f ) 0<br />
Proof:<br />
Page 12 of 72
Engineering Mathematics - II - <strong>Vector</strong> Calculus - 2007<br />
Page 13 of 72<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
k<br />
y<br />
f<br />
x<br />
f<br />
i<br />
x<br />
f<br />
z<br />
f<br />
i<br />
z<br />
f<br />
y<br />
f<br />
f<br />
Curl<br />
1<br />
2<br />
3<br />
1<br />
2<br />
3<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
y<br />
f<br />
x<br />
f<br />
x<br />
x<br />
f<br />
z<br />
f<br />
z<br />
z<br />
f<br />
y<br />
f<br />
x<br />
f)<br />
(curl<br />
div<br />
1<br />
2<br />
3<br />
1<br />
2<br />
3<br />
y<br />
z<br />
f<br />
x<br />
z<br />
f<br />
x<br />
y<br />
f<br />
z<br />
y<br />
f<br />
z<br />
x<br />
f<br />
y<br />
x<br />
f 1<br />
2<br />
2<br />
2<br />
3<br />
2<br />
1<br />
2<br />
2<br />
2<br />
3<br />
2<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
= 0<br />
Some important Identities<br />
(i) Divergence of a vector product:<br />
then<br />
fields<br />
vector<br />
are<br />
g<br />
<strong>and</strong><br />
f<br />
If<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
g<br />
curl<br />
.<br />
f<br />
f<br />
g .curl<br />
g)<br />
( f<br />
div<br />
Proof:<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
k<br />
f<br />
j<br />
f<br />
i<br />
f<br />
f<br />
Let 3<br />
2<br />
1<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
k<br />
g<br />
j<br />
g<br />
i<br />
g<br />
g 3<br />
2<br />
1<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
k<br />
)<br />
g<br />
f<br />
g<br />
(f<br />
j<br />
)<br />
g<br />
f<br />
g<br />
(f<br />
i<br />
)<br />
f<br />
g<br />
g<br />
(f<br />
g<br />
f 1<br />
2<br />
2<br />
1<br />
3<br />
1<br />
1<br />
3<br />
3<br />
2<br />
3<br />
2<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
2<br />
3<br />
3<br />
2 g<br />
f<br />
g<br />
f<br />
x<br />
g)<br />
div (f<br />
g)<br />
( f<br />
x<br />
.<br />
i<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
x<br />
g<br />
f<br />
g<br />
x<br />
f<br />
.<br />
i<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
f<br />
x<br />
g<br />
.<br />
i<br />
g<br />
x<br />
f<br />
.<br />
i<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
f<br />
.<br />
x<br />
g<br />
i<br />
g<br />
.<br />
x<br />
f<br />
i<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
f<br />
.<br />
g)<br />
(<br />
g<br />
.<br />
)<br />
f<br />
(<br />
<br />
<br />
<br />
<br />
<br />
<br />
g<br />
Curl<br />
.<br />
f<br />
f<br />
Curl<br />
.<br />
g<br />
(ii)<br />
Curl of Curl of a vector:
Engineering Mathematics - II - <strong>Vector</strong> Calculus - 2007<br />
<br />
<br />
Curl (Curl f ) grad (div f ) <br />
2<br />
f<br />
Proof:<br />
<br />
Let f f1<br />
i f2<br />
j f3<br />
k<br />
f3 f2 f1 f2 f2 f <br />
f i j<br />
1 <br />
k<br />
y z z1<br />
x x y <br />
<br />
Curl (Curl f )<br />
<br />
<br />
<br />
i <br />
<br />
f<br />
<br />
2<br />
y x<br />
<br />
<br />
<br />
f1<br />
<br />
<br />
y <br />
<br />
<br />
z<br />
<br />
<br />
<br />
f1<br />
z<br />
f <br />
<br />
3<br />
<br />
x <br />
<br />
<br />
2<br />
f <br />
2<br />
f <br />
2<br />
<br />
2<br />
<br />
<br />
<br />
1<br />
<br />
1 f<br />
<br />
1 f<br />
<br />
3<br />
i<br />
y<br />
x<br />
<br />
y<br />
2<br />
z<br />
2 z x <br />
<br />
<br />
<br />
2<br />
f <br />
2<br />
f <br />
2<br />
f <br />
2<br />
<br />
<br />
3<br />
<br />
2<br />
<br />
2 f<br />
<br />
1<br />
j<br />
z<br />
y<br />
<br />
z<br />
2<br />
x<br />
2 x y <br />
<br />
<br />
<br />
2<br />
<br />
2<br />
f <br />
2<br />
f f <br />
2<br />
1 3 3 f2<br />
<br />
k<br />
x<br />
z<br />
<br />
x<br />
2<br />
y<br />
2 y z <br />
<br />
adding<br />
<strong>and</strong><br />
subtracting<br />
<br />
2<br />
f1<br />
,<br />
y<br />
2<br />
<br />
2<br />
f3<br />
z<br />
2<br />
to<br />
each<br />
of<br />
the<br />
terms.<br />
<br />
<br />
<br />
<br />
i <br />
<br />
<br />
<br />
x<br />
f1<br />
<br />
x<br />
f<br />
<br />
2<br />
y<br />
f<br />
<br />
<br />
2<br />
<br />
<br />
3 f<br />
<br />
1<br />
z<br />
<br />
<br />
x<br />
2<br />
<br />
2<br />
f<br />
<br />
2<br />
y<br />
2<br />
<br />
2<br />
f <br />
<br />
<br />
1<br />
<br />
<br />
z<br />
2<br />
<br />
<br />
<br />
<br />
i (div f )<br />
x<br />
<br />
<br />
<br />
2<br />
f<br />
Note 1:<br />
<br />
(div f )<br />
<br />
<br />
<br />
2<br />
f<br />
<br />
<br />
The operator f . (f i f j f k) . <br />
1 2 3 <br />
<br />
<br />
i <br />
x<br />
<br />
<br />
<br />
f1<br />
f2<br />
f3<br />
x<br />
y<br />
z<br />
<br />
j<br />
y<br />
<br />
<br />
<br />
k <br />
z <br />
<br />
is called the Linear differential operator.<br />
Note 2: If is a scalar field<br />
Note 3:<br />
<br />
<br />
<br />
<br />
f . <br />
f1<br />
f2<br />
f<br />
<br />
3<br />
<br />
x y<br />
<br />
z<br />
Page 14 of 72
Engineering Mathematics - II - <strong>Vector</strong> Calculus - 2007<br />
Page 15 of 72<br />
field<br />
a vector<br />
is<br />
g<br />
If<br />
<br />
z<br />
g<br />
f<br />
y<br />
g<br />
f<br />
x<br />
g<br />
f<br />
g<br />
.<br />
f 3<br />
2<br />
1<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
(iii)<br />
<strong>Gradient</strong> of a scalar field:<br />
then<br />
fields<br />
vector<br />
are<br />
g<br />
<strong>and</strong><br />
f<br />
If<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
f<br />
)<br />
( g .<br />
g<br />
)<br />
.<br />
( f<br />
f<br />
Curl<br />
g<br />
g<br />
Curl<br />
f<br />
g)<br />
.<br />
( f<br />
Proof:<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
)<br />
g<br />
.<br />
( f<br />
x<br />
i<br />
g)<br />
.<br />
( f<br />
Consider<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
x<br />
g<br />
.<br />
f<br />
g<br />
.<br />
x<br />
f<br />
i<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
...(1)<br />
x<br />
g<br />
.<br />
f<br />
i<br />
x<br />
f<br />
g .<br />
i<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
x<br />
f<br />
i<br />
g<br />
f<br />
Curl<br />
g<br />
Now<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
x<br />
f<br />
i<br />
g<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
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x<br />
f<br />
i<br />
g .<br />
i<br />
x<br />
f<br />
g .<br />
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<br />
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<br />
<br />
f<br />
g .<br />
x<br />
f<br />
g .<br />
i<br />
<br />
<br />
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<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
...(2)<br />
x<br />
f<br />
g .<br />
i<br />
g<br />
)<br />
( g .<br />
f<br />
Curl<br />
g<br />
Similarly<br />
...(3)<br />
x<br />
g<br />
.<br />
f<br />
i<br />
g<br />
)<br />
.<br />
( f<br />
g<br />
curl<br />
f
Engineering Mathematics - II - <strong>Vector</strong> Calculus - 2007<br />
Page 16 of 72<br />
From (1), (2) <strong>and</strong> (3)<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
f<br />
)<br />
( g .<br />
g<br />
)<br />
.<br />
( f<br />
f<br />
Curl<br />
g<br />
g<br />
Curl<br />
f<br />
g)<br />
.<br />
( f<br />
Note:<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
f<br />
)<br />
.<br />
2( f<br />
f<br />
Curl<br />
f<br />
2<br />
)<br />
f<br />
.<br />
( f<br />
then<br />
g<br />
f<br />
If<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
f<br />
)<br />
.<br />
( f<br />
f<br />
Curl<br />
f<br />
)<br />
f<br />
.<br />
( f<br />
grad<br />
2<br />
1<br />
or<br />
(iv) Curl of a vector product<br />
then<br />
fields<br />
vector<br />
are<br />
g<br />
<strong>and</strong><br />
f<br />
If<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
g<br />
)<br />
.<br />
( f<br />
f<br />
)<br />
( g .<br />
f<br />
g div<br />
g<br />
div<br />
f<br />
g)<br />
( f<br />
Curl<br />
Proof:<br />
g)<br />
( f<br />
x<br />
i<br />
g)<br />
( f<br />
Curl<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
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<br />
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x<br />
g<br />
f<br />
g<br />
x<br />
f<br />
i<br />
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x<br />
g<br />
f<br />
g<br />
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f<br />
i<br />
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x<br />
g<br />
)<br />
f<br />
.<br />
( i<br />
f<br />
x<br />
g<br />
i<br />
g<br />
x<br />
f<br />
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i<br />
x<br />
f<br />
g)<br />
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( i<br />
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i<br />
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i<br />
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x<br />
f<br />
g 1<br />
1<br />
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g<br />
)<br />
.<br />
f<br />
(<br />
g<br />
div<br />
f<br />
f<br />
g div<br />
f<br />
)<br />
( g .<br />
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g<br />
)<br />
.<br />
( f<br />
f<br />
)<br />
( g .<br />
f<br />
div<br />
g<br />
g<br />
div<br />
f<br />
Integration of <strong>Vector</strong> functions<br />
a<br />
be<br />
C<br />
<strong>and</strong><br />
space<br />
in<br />
R<br />
region<br />
a<br />
over<br />
defined<br />
field<br />
vector<br />
a<br />
be<br />
f<br />
Let
Engineering Mathematics - II - <strong>Vector</strong> Calculus - 2007<br />
curve in R whose equation is<br />
<br />
r x(t) i y (t) j z(t) k<br />
Let A <strong>and</strong> B be two points on Curve C corresponding to t = a <strong>and</strong> t = b,<br />
We<br />
have<br />
<br />
t<br />
<br />
<br />
d r<br />
dt<br />
<br />
d r<br />
| |<br />
dt<br />
<br />
AB<br />
<br />
f . t ds<br />
Note 1:<br />
Note 2:<br />
Note 3:<br />
Note 4:<br />
The<br />
is<br />
<br />
Since t<br />
called the line int egral<br />
above int egral<br />
<br />
d r<br />
<br />
ds<br />
<br />
of f<br />
is also represented as<br />
<br />
C<br />
<br />
f . t ds<br />
<br />
<br />
C<br />
along curve C between<br />
<br />
f .d r<br />
<br />
<br />
f .d r <br />
<br />
f1 dx <br />
<br />
f2<br />
dy <br />
<br />
f3<br />
dz<br />
C C C C<br />
Also<br />
<br />
<br />
b<br />
dx dy dz<br />
1 dt<br />
dt dt dt<br />
a<br />
<br />
<br />
<br />
f .d r <br />
f<br />
f2<br />
f3<br />
<br />
<br />
C<br />
<br />
C<br />
<br />
f . t ds<br />
A<br />
<strong>and</strong> B.<br />
Physical Meaning:<br />
Page 17 of 72
C<br />
Engineering Mathematics - II - <strong>Vector</strong> Calculus - 2007<br />
<br />
f .d r represents the work done in moving a particle of unit<br />
along the Curve C from A to B.<br />
mass<br />
Surface Integral of a vector function<br />
The<br />
surface int egral<br />
of<br />
<br />
f<br />
over a<br />
surface<br />
S<br />
is<br />
defined<br />
as<br />
<br />
S<br />
<br />
f .n<br />
dS<br />
where<br />
<br />
n<br />
is the unit normal to the surface S <strong>and</strong> dS = dx dy<br />
Physical Meaning:<br />
The<br />
surface int egral<br />
of<br />
<br />
f<br />
gives<br />
the<br />
total normal<br />
flux<br />
through<br />
a<br />
surface.<br />
Volume integral of a vector function<br />
The<br />
volume int egral<br />
<br />
of f<br />
over<br />
a<br />
volume V<br />
is<br />
defined<br />
as<br />
<br />
V<br />
<br />
f dV<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
f1d V<br />
<br />
i<br />
<br />
f2<br />
dV<br />
<br />
j<br />
<br />
d V<br />
<br />
k<br />
V V V <br />
Integral Theorems<br />
Green's Theorem (Statement only)<br />
Let M(x,y) <strong>and</strong> N(x,y) be two functions defined in a region A in the xy plane with<br />
a simple closed curve C as its boundary then<br />
<br />
C<br />
(M dx<br />
<br />
Mdy)<br />
<br />
N<br />
<br />
<br />
x<br />
A<br />
M <br />
dx dy<br />
y <br />
Strokes Theorem (Statement only)<br />
Let S be an open surface bounded by a simple closed curve C for a field<br />
Page 18 of 72
f<br />
defined<br />
Engineering Mathematics - II - <strong>Vector</strong> Calculus - 2007<br />
over a region containing S <strong>and</strong> C.<br />
<br />
C<br />
<br />
f .d r<br />
<br />
<br />
S<br />
(Curl<br />
<br />
f ) .n<br />
dS<br />
Gauss Divergence theorem (Statement only)<br />
Let S be the closed boundary surface of a region of Volume V. Then for a<br />
<br />
vector field f defined in V <strong>and</strong> on S.<br />
<br />
S<br />
<br />
f .n<br />
dS <br />
<br />
S<br />
i.e., in Cartesian form<br />
<br />
div f dV<br />
f<br />
f<br />
f<br />
1 dx dy dz<br />
x<br />
y<br />
z<br />
V<br />
<br />
f dy dz f <br />
1<br />
<br />
2<br />
<br />
3<br />
2 dz dx f3<br />
dx dy<br />
<br />
S<br />
Note:<br />
Page 19 of 72
For<br />
Engineering Mathematics - II - <strong>Vector</strong> Calculus - 2007<br />
<br />
any vector field f , <strong>and</strong> any closed surface S<br />
<br />
S<br />
<br />
Curl f . n<br />
dS<br />
<br />
0<br />
Page 20 of 72
Engineering Mathematics - II - <strong>Vector</strong> Calculus - 2007<br />
Summary<br />
<strong>Vector</strong> function: A vector function of a scalar variable t is of the form<br />
<br />
f1(t)<br />
i f2<br />
(t) j<br />
f3(t)<br />
k<br />
<strong>Differentiation</strong> of a <strong>Vector</strong>:<br />
<br />
d f df <br />
<br />
1 df df<br />
i <br />
2<br />
j<br />
3<br />
k<br />
dt dt dt dt<br />
Equation of a space curve: Equation of a space curve is represented by<br />
<br />
r (t)<br />
<br />
x(t) i y(t) j z (t) k<br />
<br />
d r<br />
<strong>Velocity</strong> :<br />
dt<br />
space curve.<br />
is<br />
the<br />
<br />
velocity v<br />
<strong>and</strong> is<br />
along<br />
the<br />
tan gent<br />
to<br />
the<br />
Unit<br />
tan gent vector<br />
<br />
: t<br />
<br />
<br />
d r<br />
dt<br />
<br />
d r<br />
| |<br />
dt<br />
<br />
Unit normal : n <br />
|<br />
<br />
d t<br />
ds<br />
<br />
d t<br />
|<br />
ds<br />
is called the unit normal to the curve.<br />
Scalar field: A scalar corresponding to each (x,y,z) in a region R.<br />
<strong>Vector</strong> field: A vector corresponding to each (x,y,z) in a region R<br />
<br />
<br />
<br />
<br />
r f1<br />
(x, y, z) i f2<br />
(x, y, z) j f3(x, y, z) k<br />
<strong>Gradient</strong>: If<br />
<br />
is<br />
a scalar field<br />
then<br />
<br />
<br />
<br />
i<br />
x<br />
<br />
j<br />
y<br />
<br />
k<br />
z<br />
is<br />
called<br />
the<br />
gradient of .<br />
Unit Normal to the surface : If = c is a surface then<br />
<br />
| |<br />
<br />
<br />
n<br />
is<br />
the unit<br />
normal<br />
to the surface.<br />
Page 21 of 72
Engineering Mathematics - II - <strong>Vector</strong> Calculus - 2007<br />
<br />
Directional derivative: . a is called the directional derivative in<br />
<br />
the direction of a .<br />
<br />
Divergence of a vector field :If f f1<br />
i f2<br />
j f3<br />
k<br />
is<br />
a vector<br />
field<br />
then<br />
div<br />
<br />
f .<br />
f<br />
f<br />
<br />
1<br />
x<br />
f<br />
<br />
2<br />
y<br />
<br />
f3<br />
z<br />
<br />
f1<br />
x<br />
Solenoidal:<br />
<br />
f<br />
is<br />
solenoida.l of<br />
<br />
div f<br />
<br />
0<br />
Laplacian of a scalar field: Let be a scalar field then<br />
<br />
2<br />
<br />
<br />
<br />
2<br />
<br />
x<br />
2<br />
<br />
2<br />
<br />
<br />
y<br />
2<br />
<br />
2<br />
<br />
<br />
z<br />
2<br />
Curl<br />
of<br />
a<br />
vector<br />
field: Let<br />
<br />
f<br />
be<br />
a<br />
vector<br />
field<br />
then<br />
curl<br />
of<br />
<br />
f<br />
Curl<br />
<br />
f<br />
<br />
<br />
<br />
f<br />
<br />
<br />
i j k<br />
<br />
x<br />
y<br />
z<br />
f1 f2<br />
f3<br />
<br />
<br />
f <br />
<br />
3 f<br />
<br />
2<br />
i<br />
y z <br />
<br />
Irrational: f<br />
is<br />
irrational if<br />
Curl<br />
<br />
f 0<br />
Introduction<br />
In this chapter the basic concepts of Differential Calculus of scalar functions are<br />
extended to vector functions.<br />
Here we study vector differentiation, gradient, divergence, curl, solenoidal <strong>and</strong><br />
irrational fields.<br />
Also we study about vector integration <strong>and</strong> verification of the integral theorems.<br />
<strong>Vector</strong> Function of a Scalar Variable<br />
<br />
Let f f1<br />
i f2<br />
j f3<br />
k where f1, f2<br />
<strong>and</strong> f3<br />
are<br />
functions<br />
of<br />
a<br />
var iable<br />
t<br />
is called a vector function.<br />
Page 22 of 72
Engineering Mathematics - II - <strong>Vector</strong> Calculus - 2007<br />
If f1, f2 <strong>and</strong> f3 are differentiable then we define<br />
<br />
d f f(t t) f(t)<br />
df <br />
1 df <br />
2 df <br />
lim i j <br />
3<br />
k<br />
dt t0<br />
t dt dt dt<br />
Similarly higher order derivatives can also be defined.<br />
Note:<br />
<br />
d f <br />
0<br />
dt<br />
iff<br />
<br />
f<br />
is<br />
a<br />
cons tan t<br />
vector .<br />
Rules of differentiation<br />
1. If<br />
g(t) is<br />
a<br />
scalar<br />
function<br />
of<br />
t<br />
<strong>and</strong><br />
<br />
f (t) is a<br />
vector<br />
function<br />
of<br />
<br />
d d f d <br />
scalar t then ( f) f<br />
dt dt dt<br />
2. If<br />
<br />
<strong>and</strong><br />
<br />
are<br />
scalar<br />
cons tan ts,<br />
<br />
f<br />
<strong>and</strong><br />
<br />
g<br />
are vector<br />
functions of<br />
t<br />
then<br />
d<br />
dt<br />
<br />
( f<br />
<br />
<br />
g)<br />
<br />
<br />
d f<br />
<br />
dt<br />
<br />
<br />
d g<br />
<br />
dt<br />
3.<br />
d<br />
dt<br />
Proof:<br />
<br />
( f . g)<br />
<br />
d g<br />
f .<br />
dt<br />
<br />
d f <br />
. g<br />
dt<br />
<br />
Let f f1<br />
i f2<br />
j f3<br />
k<br />
<br />
f . g<br />
<br />
g g1<br />
i g2<br />
j g3<br />
k<br />
f1 g1<br />
f2<br />
g2<br />
f3<br />
g3<br />
d dg1<br />
df1<br />
dg2<br />
df2<br />
dg3<br />
df3<br />
( f . g) f1 g1<br />
f2<br />
g2<br />
f3<br />
g3<br />
dt<br />
dt dt dt dt dt dt<br />
dg<br />
<br />
<br />
<br />
1 dg<br />
<br />
2 dg<br />
<br />
3 df<br />
<br />
1 df<br />
<br />
2 df<br />
f<br />
<br />
3<br />
1 f2<br />
f3<br />
g1<br />
g2<br />
g3<br />
<br />
dt dt dt dt dt dt <br />
<br />
d g<br />
f .<br />
dt<br />
<br />
df <br />
. g<br />
dt<br />
Page 23 of 72
Engineering Mathematics - II - <strong>Vector</strong> Calculus - 2007<br />
Page 24 of 72<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
g<br />
dt<br />
f<br />
d<br />
dt<br />
d g<br />
f<br />
g)<br />
( f<br />
dt<br />
d<br />
4.<br />
Proof:<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
k<br />
f<br />
j<br />
f<br />
i<br />
f<br />
f<br />
Let 3<br />
2<br />
1<br />
<br />
<br />
<br />
<br />
<br />
<br />
k<br />
g<br />
j<br />
g<br />
i<br />
g<br />
g 3<br />
2<br />
1<br />
3<br />
2<br />
1<br />
3<br />
2<br />
1<br />
g<br />
g<br />
g<br />
f<br />
f<br />
f<br />
k<br />
j<br />
i<br />
g<br />
f<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
)<br />
f<br />
g<br />
g<br />
(f<br />
i<br />
g<br />
f 3<br />
2<br />
3<br />
2<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
)<br />
f<br />
g<br />
g<br />
(f<br />
i<br />
dt<br />
d<br />
g)<br />
( f<br />
dt<br />
d<br />
3<br />
2<br />
3<br />
2<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
dt<br />
df<br />
g<br />
f<br />
dt<br />
dg<br />
dt<br />
dg<br />
f<br />
g<br />
dt<br />
d f<br />
i<br />
3<br />
2<br />
3<br />
2<br />
3<br />
2<br />
3<br />
2<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
dt<br />
df<br />
g<br />
dt<br />
dg<br />
f<br />
i<br />
f<br />
dt<br />
dg<br />
g<br />
dt<br />
d f<br />
i<br />
3<br />
2<br />
3<br />
2<br />
3<br />
2<br />
3<br />
2<br />
dt<br />
d g<br />
f<br />
g<br />
dt<br />
f<br />
d<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
dt<br />
d h<br />
g ,<br />
,<br />
f<br />
,h<br />
dt<br />
d g<br />
,<br />
f<br />
h<br />
g,<br />
,<br />
dt<br />
f<br />
d<br />
h<br />
g,<br />
,<br />
f<br />
dt<br />
d<br />
5.<br />
Proof:<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
h<br />
.<br />
g<br />
f<br />
dt<br />
d<br />
h<br />
g,<br />
,<br />
f<br />
dt<br />
d<br />
dt<br />
d h<br />
.<br />
g<br />
f<br />
h<br />
.<br />
g<br />
f<br />
dt<br />
d<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
dt<br />
d h<br />
.<br />
g<br />
f<br />
h<br />
.<br />
dt<br />
d g<br />
f<br />
h<br />
.<br />
g<br />
dt<br />
f<br />
d<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
dt<br />
d h<br />
g,<br />
,<br />
f<br />
h<br />
,<br />
dt<br />
d g<br />
,<br />
f<br />
h<br />
g ,<br />
,<br />
dt<br />
f<br />
d
Engineering Mathematics - II - <strong>Vector</strong> Calculus - 2007<br />
<br />
<br />
d f<br />
6. If | f | cons tan t then f . 0<br />
dt<br />
Proof:<br />
<br />
Let | f | <br />
C<br />
<br />
| f | a<br />
2<br />
where a is constant<br />
<br />
<br />
f .<br />
<br />
f<br />
<br />
cons tan t<br />
<br />
d<br />
dt<br />
<br />
f .<br />
<br />
<br />
<br />
f <br />
<br />
<br />
<br />
0<br />
<br />
f .<br />
<br />
d f d f <br />
. f <br />
dt dt<br />
0<br />
<br />
<br />
d f<br />
f .<br />
dt<br />
<br />
0<br />
7. If<br />
<br />
f<br />
has<br />
a<br />
fixed<br />
direction<br />
then<br />
<br />
f<br />
<br />
d f <br />
0<br />
dt<br />
Proof:<br />
Let<br />
<br />
f<br />
<br />
<br />
f e<br />
Now<br />
<br />
f <br />
<br />
d f<br />
dt<br />
<br />
f e <br />
d<br />
dt<br />
<br />
(f e)<br />
<br />
df d e <br />
f e e 0<br />
dt dt<br />
f<br />
df<br />
dt<br />
<br />
(e e)<br />
df <br />
f (0)<br />
dt<br />
<br />
0<br />
8.If<br />
<br />
d f<br />
f <br />
dt<br />
<br />
<br />
0<br />
then<br />
<br />
f<br />
has<br />
a<br />
fixed<br />
direction.<br />
Page 25 of 72
Engineering Mathematics - II - <strong>Vector</strong> Calculus - 2007<br />
Equation of a Space Curve<br />
Let P (x,y,z) be any point in space then its position vector is<br />
<br />
OP<br />
<br />
<br />
r<br />
<br />
<br />
x i<br />
<br />
y j<br />
<br />
<br />
z k<br />
If x = x(t), y = y(t) <strong>and</strong> z = z(t) are functions of a scalar variable<br />
t then<br />
<br />
r r (t)<br />
describes<br />
a<br />
curve<br />
in<br />
space<br />
called<br />
the<br />
space<br />
curve.<br />
Note 1:<br />
<br />
d r<br />
dt<br />
represents<br />
the<br />
velocity<br />
<br />
v<br />
(rate<br />
of<br />
change<br />
of<br />
position<br />
with<br />
which<br />
the<br />
ter min al po int<br />
of<br />
<br />
r<br />
describes the<br />
curve )<br />
Note 2:<br />
<br />
d r<br />
dt<br />
is<br />
along<br />
the<br />
direction<br />
of<br />
the<br />
tan gent<br />
to<br />
the<br />
space<br />
curve<br />
at<br />
P (x, y, z)<br />
Note 3:<br />
The<br />
unit<br />
tan gent<br />
vector<br />
to<br />
the<br />
space<br />
curve<br />
is<br />
denoted by<br />
<br />
t<br />
<strong>and</strong><br />
is given by<br />
<br />
d r<br />
<br />
t <br />
dt<br />
<br />
d r<br />
| |<br />
dt<br />
Page 26 of 72
Note 4:<br />
<br />
d v<br />
dt<br />
Engineering Mathematics - II - <strong>Vector</strong> Calculus - 2007<br />
<br />
d<br />
2<br />
r<br />
<br />
is called the acceleration a of<br />
dt<br />
2<br />
the<br />
curve<br />
at time<br />
t.<br />
Unit Normal to a Space Curve<br />
Let A be a fixed point on the curve <strong>and</strong> s be the length of the arc AP where<br />
P(x,y,z) is a point on the space curve.<br />
Let<br />
<br />
t<br />
be<br />
the<br />
unit<br />
tan gent<br />
vector<br />
at<br />
P.<br />
Consider<br />
d.w.r.t s<br />
<br />
t . t<br />
1<br />
<br />
t<br />
.<br />
<br />
d t<br />
ds<br />
<br />
<br />
d t <br />
. t<br />
ds<br />
<br />
0<br />
<br />
<br />
d t<br />
2 t . <br />
ds<br />
0<br />
<br />
<br />
t<br />
<br />
d t<br />
.<br />
ds<br />
<br />
0<br />
<br />
d t<br />
ds<br />
is<br />
a normal<br />
to<br />
the<br />
space<br />
curve<br />
<strong>and</strong><br />
<br />
n<br />
<br />
<br />
d t<br />
ds<br />
<br />
d t<br />
| |<br />
ds<br />
is called<br />
the<br />
unit<br />
normal to<br />
the<br />
curve.<br />
Page 27 of 72
Engineering Mathematics - II - <strong>Vector</strong> Calculus - 2007<br />
Scalar <strong>and</strong> <strong>Vector</strong> Fields<br />
Scalar Valued Function<br />
Let P(x,y,z) which is a scalar is called a scalar point function.<br />
Note: A scalar point function is also called a scalar fixed . = (x,y,z)<br />
Ex.: (x,y,z) = x 2 + y 2 + z 2<br />
<strong>Vector</strong> Valued Function<br />
Let P(x,y,z) be a point. A vector point function is a vector whose<br />
components are real valued functions of x,y,z. A vector point function is called a<br />
<br />
vector field F .<br />
Ex:<br />
<br />
<br />
f f (x, y, z) f1<br />
i f2<br />
j f3<br />
k<br />
<br />
F<br />
<br />
<br />
xyz i<br />
<br />
<br />
x<br />
2<br />
y j<br />
<br />
<br />
yz k<br />
Differential<br />
operator<br />
:<br />
<br />
denotes<br />
the<br />
operation<br />
<br />
i<br />
x<br />
<br />
<br />
y<br />
<br />
j<br />
<br />
<br />
k<br />
z<br />
<br />
<br />
<br />
i<br />
x<br />
<br />
does<br />
not<br />
represent<br />
a<br />
vector<br />
if<br />
only<br />
defines<br />
the<br />
differential<br />
operator<br />
<strong>Gradient</strong> of a Scalar Field<br />
<br />
Let(x, y, z) be a scalar field, then i k i<br />
x z x<br />
is called the gradient of denoted by .<br />
The<br />
following<br />
are<br />
some<br />
results<br />
obtained<br />
by<br />
the<br />
definition<br />
of<br />
<br />
.<br />
1. ( )<br />
,<br />
is a scalar cons tan t.<br />
2. ( )<br />
,<br />
<strong>and</strong> are scalar fields.<br />
3. ( )<br />
,<br />
<strong>and</strong> are scalar cons tan ts<br />
4.() <br />
<br />
<br />
5. <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
2<br />
Page 28 of 72
Engineering Mathematics - II - <strong>Vector</strong> Calculus - 2007<br />
<br />
6. d dx dy dz<br />
x y z<br />
<br />
i<br />
<br />
<br />
<br />
x<br />
<br />
<br />
j<br />
<br />
y<br />
<br />
<br />
k<br />
<br />
<br />
z <br />
<br />
<br />
i<br />
dx<br />
<br />
<br />
<br />
<br />
j dy<br />
<br />
<br />
k dz<br />
<br />
<br />
d<br />
<br />
<br />
<br />
. d r<br />
Geometrical Interpretation<br />
Q (x, y, z)<br />
<br />
c<br />
represents<br />
a<br />
surface<br />
in space<br />
<strong>and</strong><br />
<br />
represents<br />
a normal<br />
to the surface at any point.<br />
Note 1: Since = c on the surface.<br />
<br />
d . d r<br />
<br />
0<br />
which<br />
shows<br />
that<br />
<br />
is<br />
perpendicular<br />
to<br />
the<br />
tan gent<br />
plane<br />
at<br />
any<br />
po int .<br />
Note<br />
2 : Unit<br />
vector<br />
along<br />
<br />
is<br />
denoted by<br />
<br />
n<br />
<br />
<br />
is<br />
| |<br />
called<br />
the<br />
unit<br />
normal<br />
to the surface (x,y,z) = c.<br />
Directional Derivative<br />
Let<br />
<br />
a<br />
be<br />
a<br />
vector<br />
inclined at<br />
an<br />
angle<br />
<br />
with<br />
<br />
then<br />
<br />
.<br />
<br />
a is<br />
called<br />
the<br />
directional<br />
derivative<br />
along<br />
<br />
a .<br />
Page 29 of 72
Engineering Mathematics - II - <strong>Vector</strong> Calculus - 2007<br />
Note : Maximum value<br />
of directional<br />
derivative<br />
is<br />
<br />
<br />
<br />
n<br />
| |<br />
Divergence of a <strong>Vector</strong> Field<br />
<br />
Let f f1<br />
i f2<br />
j f3<br />
k<br />
be<br />
a<br />
vector<br />
field<br />
then<br />
divergence<br />
of<br />
<br />
f<br />
denoted by<br />
div<br />
<br />
f<br />
or<br />
<br />
.<br />
f<br />
is<br />
defined<br />
as<br />
f<br />
f<br />
1 f2<br />
f <br />
3 i<br />
div. f . f <br />
x y z x<br />
Physical Interpretation<br />
If<br />
<br />
v the<br />
velocity<br />
of<br />
a moving<br />
fluid<br />
at<br />
a po int<br />
P<br />
at<br />
time<br />
t,<br />
then<br />
<br />
div v<br />
represents<br />
the rate at which the fluid moves out of a unit volume enclosing the point P.<br />
Solenoidal <strong>Vector</strong><br />
A<br />
vector<br />
field<br />
<br />
f<br />
is<br />
said<br />
to<br />
be<br />
solenoidal<br />
if<br />
<br />
div. f<br />
<br />
0.<br />
Properties<br />
<br />
1. div ( f g) div f div g , where f <strong>and</strong> g are vector fields.<br />
<br />
2. div ( f ) div f where is a scalar cons tan t.<br />
3.<br />
<br />
div ( f g)<br />
cons tan ts.<br />
<br />
<br />
div f <br />
<br />
div g<br />
where<br />
<br />
<strong>and</strong><br />
<br />
are<br />
scalar<br />
4.<br />
If<br />
<br />
<br />
is a scalar field <strong>and</strong> f is a vector field then div ( f ) div f .<br />
Proof:<br />
Page 30 of 72
Engineering Mathematics - II - <strong>Vector</strong> Calculus - 2007<br />
<br />
Let f f1<br />
i f2<br />
j f3<br />
k<br />
<br />
then f f1<br />
i f2<br />
j f3<br />
k<br />
<br />
div ( f ) ( f1 ) ( f2<br />
) ( f3)<br />
x y z<br />
<br />
f1<br />
x<br />
<br />
<br />
f1<br />
x<br />
<br />
f<br />
<br />
2<br />
y<br />
<br />
f2<br />
<br />
y<br />
<br />
f<br />
<br />
3<br />
y<br />
<br />
f3<br />
<br />
y<br />
<br />
<br />
<br />
<br />
<br />
f1<br />
<br />
<br />
x <br />
<br />
<br />
<br />
i<br />
x<br />
<br />
. f1<br />
i<br />
<br />
div<br />
<br />
f<br />
<br />
<br />
.<br />
f<br />
Laplacian of a scalar field <br />
Let<br />
then<br />
= (x,y,z) be a scalar field<br />
<br />
is a vector field<br />
<strong>and</strong><br />
div<br />
( )<br />
.<br />
<br />
<br />
<br />
x<br />
<br />
<br />
x <br />
<br />
<br />
<br />
y<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
y <br />
<br />
<br />
z<br />
<br />
<br />
z <br />
<br />
2<br />
<br />
<br />
x<br />
2<br />
<br />
<br />
2<br />
<br />
y<br />
2<br />
<br />
<br />
z<br />
2<br />
denoted by<br />
<br />
2 <br />
is<br />
called<br />
the<br />
Laplacian<br />
of<br />
a<br />
scalar<br />
field.<br />
Note 1:<br />
<br />
2<br />
<br />
<br />
2<br />
x<br />
2<br />
<br />
2<br />
<br />
y<br />
2<br />
<br />
2<br />
<br />
z<br />
2<br />
Note 2:<br />
A<br />
scalar<br />
field<br />
<br />
is<br />
called<br />
a harmonic<br />
function if<br />
<br />
2<br />
<br />
<br />
0<br />
Note 3:<br />
<br />
2<br />
() <br />
2<br />
, where <br />
2<br />
<strong>and</strong> are cons tan ts,<br />
<br />
<strong>and</strong><br />
<br />
are scalar fields.<br />
Curl of a <strong>Vector</strong> field<br />
<br />
Let f f1<br />
i f2<br />
j f3<br />
k<br />
then<br />
curl<br />
of<br />
<br />
f<br />
denoted by<br />
curl<br />
<br />
f<br />
or<br />
Page 31 of 72
Engineering Mathematics - II - <strong>Vector</strong> Calculus - 2007<br />
<br />
<br />
<br />
i j k<br />
f is defined as f / x / y<br />
/ z<br />
f1 f2<br />
f3<br />
<br />
<br />
f <br />
<br />
3 f<br />
<br />
2<br />
i<br />
y z <br />
Physical Interpretation<br />
<br />
If v is the velocity of a particle in rigid body rotating about a fixed<br />
axis<br />
then<br />
the<br />
angular<br />
velocity<br />
at<br />
any<br />
po int<br />
is<br />
given by<br />
1<br />
2<br />
curl<br />
<br />
v .<br />
Irrational <strong>Vector</strong><br />
<br />
f<br />
is<br />
said<br />
to<br />
irrational if<br />
curl<br />
<br />
f<br />
<br />
0<br />
Properties<br />
(i)<br />
If<br />
<br />
f<br />
<strong>and</strong><br />
<br />
g<br />
are<br />
vector<br />
fields then<br />
<br />
curl ( f<br />
<br />
<br />
g)<br />
<br />
curl<br />
<br />
f<br />
<br />
curl<br />
<br />
g<br />
<br />
<br />
(ii) Curl ( f ) Curl f , where is a scalar cons tan t.<br />
(iii)<br />
If<br />
<br />
is<br />
a<br />
scalar<br />
field<br />
<strong>and</strong><br />
<br />
f<br />
is<br />
vector<br />
field<br />
then<br />
<br />
curl ( f )<br />
<br />
<br />
curl<br />
<br />
f<br />
<br />
<br />
<br />
<br />
f<br />
Proof:<br />
<br />
<br />
Curl ( f ) ( f3)<br />
( f2<br />
) i<br />
<br />
y z <br />
<br />
<br />
<br />
f<br />
<br />
<br />
3<br />
y<br />
<br />
f <br />
2<br />
i<br />
z <br />
<br />
<br />
f3<br />
y<br />
<br />
<br />
f2<br />
i <br />
z <br />
<br />
<br />
f f <br />
<br />
<br />
Curl ( f ) <br />
3<br />
<br />
2<br />
i f3<br />
f2<br />
i ...(1)<br />
y z y z <br />
Consider<br />
<br />
<br />
f <br />
<br />
i j k<br />
/ x / y / z<br />
f1 f2<br />
f3<br />
Page 32 of 72
Engineering Mathematics - II - <strong>Vector</strong> Calculus - 2007<br />
Page 33 of 72<br />
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<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
i<br />
f<br />
z<br />
f<br />
y<br />
2<br />
3<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
f<br />
f<br />
curl<br />
)<br />
f<br />
(<br />
Curl<br />
(iv)<br />
<br />
<br />
<br />
0<br />
)<br />
(<br />
Curl<br />
Proof:<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
i<br />
y<br />
z<br />
z<br />
y<br />
)<br />
(<br />
Curl<br />
<br />
0<br />
(v). 0<br />
)<br />
f<br />
(Curl<br />
div<br />
then<br />
field<br />
vector<br />
a<br />
is<br />
f<br />
If<br />
<br />
<br />
<br />
Proof:<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
k<br />
y<br />
f<br />
x<br />
f<br />
i<br />
x<br />
f<br />
z<br />
f<br />
i<br />
z<br />
f<br />
y<br />
f<br />
f<br />
Curl<br />
1<br />
2<br />
3<br />
1<br />
2<br />
3<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
y<br />
f<br />
x<br />
f<br />
x<br />
x<br />
f<br />
z<br />
f<br />
z<br />
z<br />
f<br />
y<br />
f<br />
x<br />
f )<br />
(curl<br />
div<br />
1<br />
2<br />
3<br />
1<br />
2<br />
3<br />
y<br />
z<br />
f<br />
x<br />
z<br />
f<br />
x<br />
y<br />
f<br />
z<br />
y<br />
f<br />
z<br />
x<br />
f<br />
y<br />
x<br />
f 1<br />
2<br />
2<br />
2<br />
3<br />
2<br />
1<br />
2<br />
2<br />
2<br />
3<br />
2<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
= 0<br />
Some important Identities<br />
(i) Divergence of a vector product:<br />
then<br />
fields<br />
vector<br />
are<br />
g<br />
<strong>and</strong><br />
f<br />
If<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
g<br />
curl<br />
.<br />
f<br />
f<br />
g .curl<br />
g)<br />
( f<br />
div<br />
Proof:<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
k<br />
f<br />
j<br />
f<br />
i<br />
f<br />
f<br />
Let 3<br />
2<br />
1<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
k<br />
g<br />
j<br />
g<br />
i<br />
g<br />
g 3<br />
2<br />
1<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
k<br />
)<br />
g<br />
f<br />
g<br />
(f<br />
j<br />
)<br />
g<br />
f<br />
g<br />
(f<br />
i<br />
)<br />
f<br />
g<br />
g<br />
(f<br />
g<br />
f 1<br />
2<br />
2<br />
1<br />
3<br />
1<br />
1<br />
3<br />
3<br />
2<br />
3<br />
2<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
2<br />
3<br />
3<br />
2 g<br />
f<br />
g<br />
f<br />
x<br />
g)<br />
div (f
Engineering Mathematics - II - <strong>Vector</strong> Calculus - 2007<br />
<br />
i . ( f g)<br />
x<br />
<br />
<br />
<br />
i<br />
.<br />
<br />
<br />
f<br />
<br />
x<br />
<br />
<br />
<br />
g<br />
<br />
<br />
f<br />
<br />
<br />
g <br />
<br />
x <br />
<br />
<br />
<br />
<br />
f<br />
i . <br />
x<br />
<br />
<br />
<br />
<br />
g <br />
<br />
<br />
<br />
<br />
<br />
g<br />
i . <br />
x<br />
<br />
<br />
<br />
<br />
f <br />
<br />
<br />
<br />
<br />
<br />
<br />
i<br />
<br />
<br />
<br />
<br />
f <br />
.<br />
x <br />
<br />
<br />
g<br />
<br />
<br />
<br />
<br />
i<br />
<br />
<br />
<br />
<br />
g <br />
.<br />
x <br />
<br />
<br />
f<br />
<br />
( f ) .<br />
<br />
g<br />
<br />
( <br />
<br />
g) .<br />
<br />
f<br />
<br />
g . Curl<br />
<br />
f<br />
<br />
<br />
f . Curl<br />
<br />
g<br />
(ii)<br />
Curl of Curl of a vector:<br />
<br />
Curl (Curl f )<br />
<br />
grad (div<br />
<br />
f )<br />
<br />
<br />
<br />
2<br />
f<br />
Proof:<br />
<br />
Let f f1<br />
i f2<br />
j f3<br />
k<br />
f3 f2 f1 f2 f2 f <br />
f i j<br />
1 <br />
k<br />
y z z1<br />
x x y <br />
<br />
Curl (Curl f )<br />
<br />
<br />
<br />
i <br />
<br />
f<br />
<br />
2<br />
y x<br />
<br />
<br />
<br />
f1<br />
<br />
<br />
y <br />
<br />
<br />
z<br />
<br />
<br />
<br />
f1<br />
z<br />
f <br />
<br />
3<br />
<br />
x <br />
<br />
<br />
2<br />
f <br />
2<br />
f <br />
2<br />
<br />
2<br />
<br />
<br />
<br />
1<br />
<br />
1 f<br />
<br />
1 f<br />
<br />
3<br />
i<br />
y<br />
x<br />
<br />
y<br />
2<br />
z<br />
2 z x <br />
<br />
<br />
<br />
2<br />
f <br />
2<br />
f <br />
2<br />
f <br />
2<br />
<br />
<br />
3<br />
<br />
2<br />
<br />
2 f<br />
<br />
1<br />
j<br />
z<br />
y<br />
<br />
z<br />
2<br />
x<br />
2 x y <br />
<br />
<br />
<br />
2<br />
<br />
2<br />
f <br />
2<br />
f f <br />
2<br />
1 3 3 f2<br />
<br />
k<br />
x<br />
z<br />
<br />
x<br />
2<br />
y<br />
2 y z <br />
<br />
adding<br />
<strong>and</strong><br />
subtracting<br />
<br />
2<br />
f1<br />
,<br />
y<br />
2<br />
<br />
2<br />
f3<br />
z<br />
2<br />
to<br />
each<br />
of<br />
the<br />
terms.<br />
<br />
<br />
<br />
<br />
i <br />
<br />
<br />
<br />
x<br />
f1<br />
<br />
x<br />
f<br />
<br />
2<br />
y<br />
f<br />
<br />
<br />
2<br />
<br />
<br />
3 f<br />
<br />
1<br />
z<br />
<br />
<br />
x<br />
2<br />
<br />
2<br />
f<br />
<br />
2<br />
y<br />
2<br />
<br />
2<br />
f <br />
<br />
<br />
1<br />
<br />
<br />
z<br />
2<br />
<br />
Page 34 of 72
Engineering Mathematics - II - <strong>Vector</strong> Calculus - 2007<br />
<br />
i (div f ) <br />
2<br />
f<br />
x<br />
Note 1:<br />
<br />
(div f )<br />
<br />
<br />
<br />
2<br />
f<br />
<br />
<br />
The operator f . (f i f j f k) . <br />
1 2 3 <br />
<br />
<br />
i <br />
x<br />
<br />
<br />
<br />
f1<br />
f2<br />
f3<br />
x<br />
y<br />
z<br />
<br />
j<br />
y<br />
<br />
<br />
<br />
k <br />
z <br />
<br />
is called the Linear differential operator.<br />
Note 2: If is a scalar field<br />
Note 3:<br />
If<br />
<br />
<br />
<br />
<br />
f . <br />
f1<br />
f2<br />
f<br />
<br />
3<br />
<br />
x y<br />
<br />
g is a vector field<br />
<br />
z<br />
<br />
g g<br />
f . g f1<br />
f2<br />
f3<br />
x<br />
y<br />
<br />
g<br />
z<br />
(iii)<br />
<strong>Gradient</strong> of a scalar field:<br />
<br />
If f<br />
<br />
<strong>and</strong> g are vector fields<br />
then<br />
<br />
<br />
( f . g)<br />
<br />
<br />
f<br />
<br />
Curl<br />
<br />
g<br />
<br />
<br />
g<br />
<br />
Curl<br />
<br />
f<br />
<br />
<br />
( f . )<br />
g<br />
<br />
<br />
( g . )<br />
f<br />
Proof:<br />
Consider<br />
<br />
( f . g) i ( f . g)<br />
x<br />
<br />
<br />
<br />
<br />
<br />
f g <br />
i . g f . <br />
x x <br />
<br />
<br />
<br />
<br />
<br />
<br />
f <br />
i g . <br />
x <br />
<br />
<br />
<br />
<br />
g <br />
i f . <br />
x <br />
<br />
...(1)<br />
<br />
Now g Curl<br />
<br />
f<br />
<br />
<br />
g <br />
<br />
<br />
i<br />
<br />
<br />
f<br />
x<br />
Page 35 of 72
Engineering Mathematics - II - <strong>Vector</strong> Calculus - 2007<br />
<br />
f <br />
g i <br />
x <br />
<br />
<br />
<br />
<br />
<br />
<br />
f <br />
<br />
g . i<br />
<br />
x<br />
<br />
<br />
<br />
<br />
<br />
g . i <br />
<br />
<br />
<br />
f <br />
<br />
x<br />
<br />
<br />
<br />
i<br />
<br />
<br />
f <br />
g . <br />
x<br />
<br />
<br />
<br />
<br />
g . <br />
f<br />
<br />
<br />
<br />
<br />
g<br />
<br />
Curl<br />
<br />
f<br />
<br />
<br />
( g . )<br />
g<br />
<br />
<br />
<br />
i<br />
<br />
<br />
f<br />
g .<br />
x<br />
<br />
<br />
<br />
<br />
<br />
<br />
...(2)<br />
Similarly<br />
<br />
f <br />
<br />
curl g<br />
From (1), (2) <strong>and</strong> (3)<br />
Note:<br />
<br />
( f . g)<br />
<br />
( f . )<br />
g <br />
<br />
f Curl<br />
<br />
g<br />
<br />
<br />
<br />
g <br />
i f . <br />
x <br />
<br />
<br />
g Curl<br />
<br />
f<br />
...(3)<br />
<br />
( f . )<br />
g ( g . )<br />
f<br />
If<br />
<br />
f<br />
<br />
<br />
g<br />
then<br />
<br />
<br />
( f . f )<br />
<br />
<br />
2 f<br />
<br />
Curl<br />
<br />
f 2( f . )<br />
f<br />
or<br />
1 <br />
grad ( f . f )<br />
2<br />
<br />
<br />
f <br />
Curl<br />
<br />
f <br />
<br />
( f . )<br />
f<br />
(iv) Curl of a vector product<br />
<br />
If f<br />
<strong>and</strong><br />
<br />
g<br />
are<br />
vector<br />
fields<br />
then<br />
<br />
Curl ( f <br />
<br />
g)<br />
<br />
<br />
f div<br />
<br />
g g div f ( g . )<br />
f ( f . )<br />
g<br />
Proof:<br />
<br />
Curl ( f g)<br />
<br />
<br />
<br />
i<br />
<br />
<br />
x<br />
<br />
( f g)<br />
<br />
<br />
<br />
i<br />
<br />
<br />
<br />
f<br />
<br />
x<br />
<br />
<br />
g<br />
<br />
<br />
f <br />
<br />
g <br />
<br />
x<br />
<br />
<br />
Page 36 of 72
Engineering Mathematics - II - <strong>Vector</strong> Calculus - 2007<br />
Page 37 of 72<br />
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<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
x<br />
g<br />
f<br />
g<br />
x<br />
f<br />
i<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
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<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
x<br />
g<br />
f<br />
i<br />
g<br />
x<br />
f<br />
i<br />
<br />
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<br />
<br />
<br />
<br />
<br />
x<br />
g<br />
)<br />
f<br />
.<br />
( i<br />
f<br />
x<br />
g<br />
i<br />
g<br />
x<br />
f<br />
.<br />
i<br />
x<br />
f<br />
g)<br />
.<br />
( i<br />
<br />
<br />
<br />
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<br />
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<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
x<br />
g<br />
f<br />
x<br />
g<br />
.<br />
i<br />
f<br />
x<br />
f<br />
.<br />
i<br />
g<br />
x<br />
f<br />
g 1<br />
1<br />
<br />
<br />
<br />
<br />
<br />
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<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
g<br />
)<br />
.<br />
( f<br />
g<br />
div<br />
f<br />
f<br />
g div<br />
f<br />
)<br />
( g .<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
g<br />
)<br />
.<br />
f<br />
(<br />
f<br />
)<br />
( g .<br />
f<br />
div<br />
g<br />
g<br />
div<br />
f<br />
Question <strong>and</strong> answers<br />
01. If<br />
<br />
<br />
<br />
h<br />
g,<br />
,<br />
f<br />
are vectors functions of a scalar variable t. Prove that<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
dt<br />
d h<br />
x<br />
g<br />
x<br />
f<br />
h<br />
x<br />
dt<br />
d g<br />
x<br />
f<br />
h<br />
x<br />
g<br />
x<br />
dt<br />
f<br />
d<br />
h<br />
x<br />
g<br />
x<br />
f<br />
dt<br />
d<br />
Suggested answer:<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
h<br />
x<br />
g<br />
dt<br />
d<br />
x<br />
f<br />
h<br />
x<br />
g<br />
x<br />
dt<br />
f<br />
d<br />
h<br />
x<br />
g<br />
x<br />
f<br />
dt<br />
d<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
dt<br />
d h<br />
x<br />
g<br />
h<br />
x<br />
dt<br />
d g<br />
x<br />
f<br />
h<br />
x<br />
g<br />
x<br />
dt<br />
f<br />
d<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
dt<br />
d h<br />
x<br />
g<br />
fx<br />
h<br />
x<br />
dt<br />
d g<br />
x<br />
f<br />
h<br />
x<br />
g<br />
x<br />
dt<br />
f<br />
d<br />
02. If<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
k<br />
t<br />
j<br />
sin t<br />
i<br />
cos t<br />
g<br />
2t k,<br />
j<br />
e<br />
i<br />
e<br />
f<br />
,<br />
e<br />
t<br />
t<br />
2t
d<br />
find i)<br />
dt<br />
Engineering Mathematics - II - <strong>Vector</strong> Calculus - 2007<br />
<br />
<br />
d <br />
d <br />
f , ii) f g iii) f x g <br />
<br />
<br />
dt<br />
<br />
dt<br />
<br />
Suggested answer:<br />
<br />
i) f e<br />
t<br />
i e<br />
3t<br />
j<br />
<br />
2te<br />
2t<br />
k<br />
d<br />
dt<br />
<br />
f e<br />
t<br />
<br />
<br />
<br />
i 3e<br />
3t<br />
j <br />
2<br />
<br />
<br />
2te<br />
2t<br />
<br />
<br />
e<br />
2t<br />
<br />
k<br />
<br />
ii)<br />
d<br />
dt<br />
<br />
<br />
f g (e<br />
t<br />
<br />
<br />
<br />
sin t) i ( e<br />
t<br />
<br />
cos t) j<br />
(<br />
<br />
2 1) k<br />
iii)<br />
<br />
f x g <br />
<br />
i j<br />
e<br />
t<br />
e<br />
t<br />
cos t sin t<br />
<br />
k<br />
2t<br />
t<br />
<br />
i( te<br />
t<br />
<br />
2t sin t)<br />
<br />
j(te<br />
t<br />
<br />
e<br />
t<br />
cos t) k(e<br />
t<br />
sin t e<br />
t<br />
)<br />
<br />
d <br />
( f x g) i te<br />
t<br />
e<br />
t<br />
2t cos t 2 sin t<br />
dt<br />
<br />
<br />
j e<br />
t<br />
te<br />
t<br />
e<br />
t cos t e<br />
t<br />
sin t<br />
<br />
<br />
k e<br />
t<br />
cos t e<br />
t<br />
sin t e<br />
t cos t e<br />
t<br />
sin t .<br />
<br />
<br />
<br />
03. A particle moves along the curve whose parametric coordinates are x = e -t ,<br />
y = 2 cos 3t, z = 2 sin 3t. Here t is time i) Determine its velocity <strong>and</strong><br />
acceleration at time t ii) find the magnitude of velocity <strong>and</strong> acceleration at t =<br />
0.<br />
Suggested answer:<br />
Page 38 of 72
Engineering Mathematics - II - <strong>Vector</strong> Calculus - 2007<br />
i) Equation to the curve is<br />
<br />
r e<br />
t<br />
<br />
<br />
i<br />
2 cos 3t j<br />
2 sin3t k<br />
<br />
v <br />
<br />
d r<br />
dt<br />
<br />
<br />
e<br />
t<br />
i 6 sin3t j<br />
6 cos 3t k<br />
<br />
a <br />
<br />
d<br />
2<br />
r<br />
dt<br />
2<br />
<br />
<br />
<br />
e<br />
t<br />
i 18 cos 3t j<br />
18 sin3t k<br />
ii) At t = 0,<br />
<br />
v i 6 k,<br />
<br />
a i 18 j<br />
<strong>and</strong><br />
<br />
<br />
| v | 37, | a | 325<br />
04. Find the unit tangent vectors at any point on the curve x = t 2 +1, y = 4t - 3,<br />
z = 2t 2 - 6t. Determine the unit tangent at the point t = 2.<br />
Suggested answer:<br />
Equation to the space curve is<br />
<br />
<br />
<br />
r (t<br />
2<br />
1) i<br />
(4t 3) j<br />
(2t<br />
2<br />
6t) k<br />
<br />
d r<br />
dt<br />
<br />
<br />
2t i 4 j<br />
(4t 6) k<br />
<br />
d r <br />
<br />
dt <br />
t2<br />
<br />
4 i 4 j<br />
2 k<br />
unit tangent<br />
<br />
t <br />
<br />
d r<br />
dt<br />
<br />
d r<br />
dt<br />
Page 39 of 72
Engineering Mathematics - II - <strong>Vector</strong> Calculus - 2007<br />
<br />
4 i 4 j<br />
2k<br />
<br />
36<br />
<br />
2<br />
3<br />
<br />
i <br />
2<br />
3<br />
1 <br />
j<br />
k<br />
3<br />
05. A particle moves so that its position vector is given by<br />
<br />
r cos t<br />
i<br />
sin t<br />
j,<br />
is<br />
a<br />
cons tan t<br />
show<br />
that<br />
<br />
i) velocity v<br />
is perpendicular to<br />
<br />
r<br />
ii) acceleration<br />
to the distance from the origin.<br />
<br />
a is directed towards the origin <strong>and</strong> has magnitude proportional<br />
<br />
iii) r x v is a cons tan t vector.<br />
Suggested answer:<br />
i)<br />
<br />
r cos t<br />
i sin t<br />
j<br />
<br />
d r<br />
dt<br />
<br />
<br />
sin t<br />
i cos t<br />
j v<br />
Now<br />
<br />
v . r 0<br />
<br />
v r<br />
ii)<br />
<br />
a <br />
<br />
d<br />
2<br />
r<br />
dt<br />
2<br />
<br />
<br />
<br />
2<br />
cos t<br />
i sin t<br />
j<br />
<br />
a <br />
2<br />
r<br />
<br />
| a | 2 <br />
| r |<br />
iii)<br />
<br />
r x v<br />
Page 40 of 72
Engineering Mathematics - II - <strong>Vector</strong> Calculus - 2007<br />
<br />
<br />
<br />
(cos t<br />
i sin t<br />
j)x( sin t<br />
i cos t<br />
j)<br />
2 cos t( i x j) sin<br />
2<br />
<br />
t( j x i)<br />
<br />
<br />
cos<br />
2<br />
t k<br />
sin<br />
2<br />
t<br />
k<br />
<br />
k<br />
= constant vector.<br />
<br />
<br />
d f <br />
06. Two vector f <strong>and</strong> g such that x f ,<br />
dt<br />
<br />
d <br />
for some , prove that ( f x g) x( f x g)<br />
.<br />
dt<br />
<br />
d g<br />
dt<br />
<br />
x g<br />
Suggested answer:<br />
d<br />
dt<br />
<br />
d g d f <br />
( f x g) f x x g<br />
dt dt<br />
<br />
f x( x g) ( x f )x g<br />
<br />
( f . g) ( f . )<br />
g ( g . )<br />
f ( g . f ) <br />
<br />
( g . )<br />
f ( f . )<br />
g<br />
<br />
<br />
x( f x g)<br />
07. If x = t 2 +1, y = 4t - 3, z = 2t 2 - 6t represents the parametric equations of a<br />
space curve. Find the angle between the unit tangents at t = 1 <strong>and</strong> t = 2.<br />
Page 41 of 72
Engineering Mathematics - II - <strong>Vector</strong> Calculus - 2007<br />
Suggested answer:<br />
<br />
<br />
<br />
<br />
r (t<br />
2<br />
1) i (4t 3) j<br />
(2t<br />
2<br />
6t) k<br />
<br />
d r<br />
dt<br />
<br />
<br />
2t i 4 j<br />
(4t 6) k<br />
<br />
t <br />
<br />
<br />
2t i 4 j<br />
(4t 6)k<br />
4t<br />
2<br />
16 16t<br />
2<br />
36 48t<br />
<br />
<br />
<br />
2t i 4 j<br />
(4t 6)k<br />
20t<br />
2<br />
48t 52<br />
<br />
t t1<br />
<br />
<br />
2 i 4 j<br />
2 k<br />
24<br />
<br />
a<br />
(say)<br />
<br />
t t2<br />
<br />
<br />
4 i 4 j<br />
2 k<br />
36<br />
<br />
2 2 2 <br />
i j<br />
k b (say)<br />
3 3 3<br />
Let be the angle between<br />
<br />
a<br />
<br />
<strong>and</strong> b<br />
<br />
<br />
<br />
a . b<br />
cos 1 <br />
<br />
<br />
|<br />
a || b | <br />
<br />
<br />
5<br />
cos 1 <br />
3<br />
6 <br />
08. If<br />
<br />
f cosnt <br />
sinnt<br />
<br />
<br />
where n is a constant scalar <strong>and</strong><br />
<br />
<strong>and</strong> <br />
are<br />
constant vectors prove that<br />
Page 42 of 72
Engineering Mathematics - II - <strong>Vector</strong> Calculus - 2007<br />
<br />
d f <br />
i) f x n( a x b)<br />
dt<br />
ii)<br />
<br />
d<br />
2<br />
f<br />
dt<br />
2<br />
<br />
n<br />
2<br />
f<br />
iii)<br />
<br />
<br />
<br />
f ,<br />
<br />
<br />
<br />
d f d<br />
2<br />
f <br />
, 0<br />
dt dt<br />
2 <br />
<br />
<br />
Suggested answer:<br />
Given<br />
<br />
f cos nt a sinnt b<br />
<br />
d f<br />
i) f x<br />
dt<br />
<br />
d f<br />
<br />
<br />
n sinnt a n cos nt b<br />
dt<br />
<br />
<br />
<br />
(cos nt a sinnt b )x( n sinnt a n cos nt b )<br />
<br />
<br />
n cos<br />
2<br />
nt( a x b ) n sin<br />
2<br />
t(b x a)<br />
<br />
n( a x b )<br />
<br />
d<br />
2<br />
f<br />
ii)<br />
dt<br />
2<br />
<br />
<br />
n<br />
2<br />
cos nt a n<br />
2<br />
sinnt b<br />
<br />
<br />
f<br />
<br />
<br />
<br />
iii) f ,<br />
<br />
<br />
<br />
d f<br />
,<br />
dt<br />
n 2 0<br />
<br />
<br />
d<br />
2<br />
f <br />
d f <br />
f , , n<br />
2 <br />
f<br />
dt<br />
2 <br />
dt<br />
<br />
<br />
<br />
<br />
<br />
<br />
09. A particle moves along the curve x = t 3 + 1, y = t 2 , z = t + 5 where t is the<br />
time. Find the components of velocity <strong>and</strong> acceleration along the vector<br />
Page 43 of 72
Engineering Mathematics - II - <strong>Vector</strong> Calculus - 2007<br />
<br />
c i 3 j<br />
2 k at t = 1.<br />
Suggested answer:<br />
<br />
r (t<br />
3<br />
<br />
1) i t<br />
2<br />
j<br />
(t 5) k<br />
<br />
d r <br />
3t<br />
2<br />
i 2t j<br />
k<br />
dt<br />
<br />
d<br />
2<br />
r<br />
dt<br />
2<br />
<br />
6t i 2 j<br />
<br />
d r <br />
<br />
dt <br />
t1<br />
<br />
v 3 i 2 j<br />
k<br />
Now velocity along<br />
<br />
c is<br />
<br />
v c <br />
11<br />
14<br />
12<br />
<strong>Acceleration</strong> along c is a . c .<br />
14<br />
<br />
<br />
d d A d<br />
2<br />
A d A d<br />
3<br />
<br />
<br />
<br />
A <br />
10. Show that A, , A . x .<br />
dt<br />
<br />
dt dt<br />
2 <br />
dt dt<br />
3<br />
<br />
<br />
<br />
<br />
<br />
<br />
Suggested answer:<br />
d<br />
dt<br />
<br />
<br />
<br />
A,<br />
<br />
<br />
<br />
d A<br />
,<br />
dt<br />
<br />
d<br />
2<br />
A <br />
2 <br />
dt <br />
<br />
<br />
d A<br />
,<br />
dt<br />
<br />
<br />
<br />
d A<br />
,<br />
dt<br />
<br />
<br />
d<br />
2<br />
A <br />
<br />
A,<br />
dt<br />
2<br />
<br />
<br />
<br />
d<br />
2<br />
A<br />
,<br />
dt<br />
2<br />
<br />
<br />
d<br />
2<br />
A <br />
<br />
A,<br />
dt<br />
2<br />
<br />
<br />
<br />
d A<br />
,<br />
dt<br />
<br />
d<br />
3<br />
A <br />
3 <br />
dt <br />
<br />
Page 44 of 72
Engineering Mathematics - II - <strong>Vector</strong> Calculus - 2007<br />
<br />
<br />
d A d<br />
3<br />
A <br />
0 0 A,<br />
,<br />
dt 3 <br />
dt <br />
<br />
<br />
<br />
<br />
11. If R A e<br />
nt<br />
B e<br />
nt<br />
, where A <strong>and</strong> B are cons tan t vectors<br />
<br />
d<br />
2<br />
R <br />
then show that n<br />
2<br />
R .<br />
dt<br />
2<br />
Suggested answer:<br />
<br />
d R<br />
dt<br />
<br />
ne<br />
nt<br />
A ne<br />
nt<br />
B<br />
<br />
d<br />
2<br />
R<br />
dt<br />
2<br />
<br />
n<br />
2<br />
e<br />
nt<br />
A n<br />
2<br />
e<br />
nt<br />
B<br />
<br />
n 2 R<br />
<br />
12. Find unit vector normal to the curve 4 sin t i 4 cos t j<br />
3t k .<br />
Suggested answer:<br />
We have<br />
<br />
r 4 sin t i 4 cos t j<br />
3t k<br />
<br />
d r<br />
dt<br />
<br />
4 cos t i 4 sin t j<br />
3k<br />
<br />
d r<br />
dt<br />
5<br />
Page 45 of 72
t <br />
<br />
d r<br />
dt<br />
<br />
d r<br />
dt<br />
Engineering Mathematics - II - <strong>Vector</strong> Calculus - 2007<br />
<br />
<br />
4 cos t i 4 sin t j<br />
3k<br />
5<br />
<br />
d t<br />
ds<br />
<br />
<br />
d t dt <br />
<br />
dt ds <br />
<br />
d t/ dt<br />
ds / dt<br />
<br />
also<br />
ds d r<br />
<br />
dt dt<br />
5<br />
<br />
d t<br />
<br />
ds<br />
<br />
1<br />
25<br />
<br />
( 4 sin t i 4 cos t j)<br />
<strong>and</strong><br />
<br />
N <br />
<br />
4<br />
(sin t<br />
25<br />
<br />
i cos t j)<br />
16 /(25)<br />
2 <br />
<br />
<br />
(sin t i cos j)<br />
13. Find the unit normal vector to the curve x = 3 cost, y = 3sint, x = 4t.<br />
Suggested answer:<br />
We have<br />
<br />
r 3 cos t i 3 sin t j<br />
4t k<br />
<br />
d r<br />
v <br />
dt<br />
<br />
| v | 5<br />
<br />
3 sin t i 3 cos t j<br />
4 k<br />
(tan gent vector)<br />
Page 46 of 72
Engineering Mathematics - II - <strong>Vector</strong> Calculus - 2007<br />
<br />
t <br />
<br />
d r / dt<br />
ds / dt<br />
<br />
d t/ dt<br />
<br />
<br />
| d r / dt |<br />
<br />
1 <br />
( 3 cos t i 3 sin t j)<br />
25<br />
unit normal vector<br />
<br />
d t/ ds<br />
N <br />
<br />
| d t/ ds |<br />
<br />
3 <br />
(cos t i sin t j)<br />
25<br />
3 / 25<br />
<br />
(cos t i sin t j)<br />
<br />
14. The position vectors of a moving particle at time t is r t<br />
2<br />
i t<br />
3<br />
j<br />
t<br />
4<br />
k .<br />
Find the tangential <strong>and</strong> normal components of its acceleration at t = 1.<br />
Suggested answer:<br />
<br />
r t<br />
2<br />
<br />
i<br />
t<br />
3<br />
j<br />
t<br />
4<br />
k<br />
<br />
v <br />
<br />
d r<br />
dt<br />
<br />
2t i 3t<br />
2<br />
j<br />
4t<br />
4<br />
k<br />
<br />
v t1<br />
<br />
2 i 3 j<br />
4 k<br />
<br />
v <br />
1 <br />
(2 i 3 j<br />
4 k)<br />
29<br />
<br />
a t1<br />
<br />
2 i 6 j<br />
12 k<br />
Page 47 of 72
Engineering Mathematics - II - <strong>Vector</strong> Calculus - 2007<br />
tangential component acceleration is<br />
<br />
a .v <br />
70<br />
29<br />
<br />
Normal component of acceleration |<br />
a ( a .v) v |<br />
<br />
<br />
82 i 36 j<br />
68 k<br />
29<br />
<br />
1<br />
29<br />
82<br />
2<br />
36<br />
2<br />
68<br />
2<br />
= 3.8774<br />
15. If is a function of r only, then prove that<br />
1 d<br />
d<br />
<br />
r r .<br />
r dr dr<br />
Suggested answer:<br />
We have<br />
r<br />
2<br />
<br />
r . r x<br />
2<br />
y<br />
2<br />
z<br />
2<br />
r<br />
x<br />
<br />
x<br />
r<br />
,<br />
r<br />
y<br />
<br />
y<br />
r<br />
,<br />
r<br />
z<br />
<br />
z<br />
r<br />
<br />
<br />
i<br />
x<br />
<br />
j<br />
y<br />
<br />
k<br />
z<br />
d<br />
i<br />
dr<br />
x<br />
r<br />
d<br />
j<br />
dr<br />
y<br />
r<br />
d<br />
k<br />
dr<br />
z<br />
r<br />
1 d<br />
<br />
(x i y j<br />
z k)<br />
r dr<br />
Page 48 of 72
Engineering Mathematics - II - <strong>Vector</strong> Calculus - 2007<br />
<br />
1 d<br />
d<br />
r r d<br />
<br />
r r<br />
r dr dr r dr<br />
16. Find <br />
in the following<br />
i)<br />
ii)<br />
x<br />
2<br />
y<br />
3<br />
z<br />
4<br />
at (1, 0, -2)<br />
<br />
1<br />
r<br />
2<br />
Suggested answer:<br />
<br />
<br />
<br />
i) <br />
2xy<br />
3<br />
z<br />
4<br />
i 3x<br />
2<br />
y<br />
2<br />
z<br />
4<br />
j<br />
4x<br />
2<br />
y<br />
3<br />
z<br />
3<br />
k<br />
( )<br />
(1,0, 2)<br />
<br />
2 i 3 j<br />
4 k<br />
ii)<br />
<br />
1<br />
r<br />
2<br />
we have<br />
<br />
<br />
1 d<br />
<br />
r<br />
r dr<br />
<br />
1 2 <br />
<br />
r<br />
r r<br />
3<br />
<br />
<br />
<br />
2 r<br />
r<br />
4<br />
17. Find the equation of the tangent plane to the surface xyz = 6 at the point (1,<br />
2, 3).<br />
Suggested answer:<br />
xyz<br />
<br />
<br />
yz i xz j<br />
xy k<br />
<br />
(1,2,3)<br />
6 i 3 j<br />
2 k<br />
<br />
Page 49 of 72
Engineering Mathematics - II - <strong>Vector</strong> Calculus - 2007<br />
unit normal to the surface<br />
<br />
n <br />
|<br />
<br />
<br />
<br />
|<br />
<br />
6 i 3 j<br />
2 k<br />
7<br />
<br />
Equation of the tangent plane in vector form is ( r a).n 0<br />
Where<br />
<br />
r x i y j<br />
z k,<br />
<br />
a i 2 j<br />
3k<br />
ie<br />
6(x 1) 3(y 2) 2(z 3) 0<br />
i.e., 6x + 3y + 2z = 18<br />
18. Find the angle between the surfaces xlogz = y 2 - 1 <strong>and</strong> x 2 y = 2 - z at the<br />
point (1, 1, 1).<br />
Suggested answer:<br />
Let <br />
2<br />
1 x log z y<br />
2 x 2 y z<br />
1<br />
<br />
log z i 2y j<br />
x<br />
z<br />
<br />
k<br />
2<br />
<br />
2xy i x<br />
2<br />
j<br />
k<br />
( 1)<br />
(1,1,1 )<br />
<br />
2<br />
j<br />
k<br />
( 2)<br />
(1, 1, 1)<br />
<br />
2 i j<br />
k<br />
The angle between the surfaces is the angle between 1<br />
<strong>and</strong> 2.<br />
Page 50 of 72
Engineering Mathematics - II - <strong>Vector</strong> Calculus - 2007<br />
Let be the angle between 1 <strong>and</strong> <br />
2<br />
.<br />
cos<br />
1 1<br />
<br />
<br />
<br />
2<br />
| 1<br />
|| 2<br />
|<br />
cos 1 1<br />
<br />
<br />
30<br />
<br />
cos 1<br />
1<br />
30<br />
19. Find the constants a <strong>and</strong> b such that the surfaces ax 2 - byz = (a + 2)x <strong>and</strong><br />
4x 2 y+z 3 are orthogonal at (1, -1, 2).<br />
Suggested answer:<br />
Let ax<br />
2<br />
1 byz (a 2)x<br />
...(1)<br />
4x<br />
2<br />
y z<br />
3<br />
2 4<br />
...(2)<br />
1<br />
<br />
[2ax (a 2)] i bz j<br />
by k<br />
2<br />
<br />
8xy i 4x<br />
2<br />
j<br />
3z<br />
2<br />
k<br />
<br />
( 1)<br />
(1, 1,2)<br />
(a 2) i 2b j<br />
b k, ( 2)<br />
(1, 1,2)<br />
<br />
8<br />
i 4 j<br />
12 k<br />
given 1 <strong>and</strong> 2<br />
are orthogonal<br />
1. 2<br />
0<br />
i.e., -8(a - 2) + 4(-2b) + 12b = 0<br />
2a<br />
b 4<br />
...(3)<br />
Page 51 of 72
Engineering Mathematics - II - <strong>Vector</strong> Calculus - 2007<br />
Also 1 2<br />
at (1, -1, 2)<br />
a 2b (a 2) 4<br />
8 4<br />
i.e., 2b - 2 = 0<br />
b 1<br />
substituting in (3), we get<br />
a = 5/2.<br />
20. Find the directional derivative of xyz<br />
along the direction<br />
of the normal to the surface x<br />
2<br />
z y<br />
2<br />
x yz<br />
2<br />
3, at( 1,1,1).<br />
Suggested answer:<br />
Let<br />
<br />
xyz,<br />
x<br />
2<br />
z y<br />
2<br />
x yz<br />
2<br />
<br />
<br />
yz i<br />
xz j<br />
xy k<br />
<br />
at<br />
<br />
(1, 1, 1) i j<br />
k<br />
<br />
<br />
<br />
(2xz y<br />
2<br />
) i (2yx z<br />
2<br />
) j<br />
(x<br />
2<br />
<br />
2zy) k<br />
<br />
at<br />
<br />
(1, 1, 1) 3 i 3 j<br />
3k<br />
<br />
n <br />
|<br />
<br />
<br />
<br />
|<br />
<br />
i j<br />
k<br />
3<br />
Directional directive of along<br />
<br />
n is<br />
Page 52 of 72
Engineering Mathematics - II - <strong>Vector</strong> Calculus - 2007<br />
1(1) 1(1) (1)<br />
. n <br />
3<br />
3<br />
<br />
<br />
<br />
21. If F (x<br />
3<br />
y<br />
3<br />
z<br />
3<br />
3xyz) find div F <strong>and</strong> curl F .<br />
Suggested answer:<br />
<br />
F (3x<br />
2<br />
<br />
3yz) i<br />
(3y<br />
2<br />
<br />
3xz) j<br />
(3z<br />
2<br />
<br />
3xy) k<br />
<br />
div F 6x 6y 6z<br />
<br />
i j k<br />
<br />
Curl F <br />
x y z<br />
3x<br />
2<br />
3yz 3y<br />
2<br />
3xz 3z<br />
2<br />
3xy<br />
<br />
<br />
<br />
i(<br />
3x<br />
3x) j( 3y<br />
3y) k( 3z<br />
3z)<br />
<br />
0<br />
22. If<br />
<br />
is a constant vector <strong>and</strong><br />
<br />
v x r<br />
then show that<br />
<br />
div v 0.<br />
Suggested answer:<br />
Let<br />
<br />
r x i y j<br />
z k<br />
<br />
v x r<br />
Page 53 of 72
Engineering Mathematics - II - <strong>Vector</strong> Calculus - 2007<br />
<br />
i j k<br />
1 2<br />
3<br />
x y z<br />
<br />
<br />
<br />
v i[ 2z<br />
3y]<br />
j[ 1z<br />
2x]<br />
k[ 1y<br />
2x]<br />
<br />
<br />
<br />
div v ( 2z<br />
3y)<br />
( 1y<br />
g3x)<br />
( 1y<br />
2x)<br />
x<br />
y<br />
z<br />
= 0<br />
<br />
<br />
<br />
<br />
23. If f (x 2y z) i (x y z) j<br />
( x<br />
y 2z)k find div f .<br />
Suggested answer:<br />
<br />
div f <br />
<br />
<br />
<br />
(x 2y z) (x y z) ( x<br />
y 2z)<br />
x<br />
y<br />
z<br />
1 1 <br />
2<br />
2<br />
24. If<br />
<br />
f x<br />
2<br />
<br />
i<br />
y<br />
2<br />
j<br />
z k <strong>and</strong> g yz i zx j<br />
xy k<br />
show that<br />
<br />
f x g<br />
is a solenoidal vector.<br />
Suggested answer:<br />
<br />
f x g <br />
<br />
i<br />
x<br />
2<br />
yz<br />
<br />
j<br />
y<br />
2<br />
zx<br />
<br />
k<br />
z<br />
2<br />
xy<br />
<br />
f x g i{xy<br />
3<br />
<br />
<br />
xz<br />
3<br />
} j{x<br />
3<br />
y yz<br />
3<br />
} k{zx<br />
3<br />
zy<br />
3<br />
}<br />
Page 54 of 72
div<br />
Engineering Mathematics - II - <strong>Vector</strong> Calculus - 2007<br />
<br />
( f x g) (y<br />
3<br />
z<br />
3<br />
) (z<br />
3<br />
x<br />
3<br />
) (x<br />
3<br />
y<br />
3<br />
)<br />
0<br />
<br />
f x g is solenoidal.<br />
25. Prove that<br />
<br />
div (r<br />
n<br />
r ) (n 3)r<br />
n<br />
.<br />
Suggested answer:<br />
<br />
div (r<br />
n<br />
r ) r<br />
n<br />
. r r<br />
n<br />
div r<br />
<br />
nr<br />
n1<br />
<br />
x<br />
r<br />
<br />
i nr<br />
n1<br />
y<br />
r<br />
<br />
<br />
<br />
<br />
z<br />
nr<br />
n 1<br />
k.<br />
r r<br />
n<br />
<br />
r <br />
<br />
<br />
x <br />
x<br />
<br />
<br />
y <br />
y<br />
<br />
<br />
z<br />
<br />
nr<br />
n1<br />
<br />
{x i y j<br />
z k}. r r<br />
n<br />
.(3)<br />
r<br />
nr<br />
n2<br />
.r<br />
2<br />
r<br />
n<br />
(3)<br />
( n 3)r<br />
n<br />
26. If<br />
<br />
a is a constant vector then prove that<br />
<br />
i) div( a x r ) 0<br />
<br />
ii) div ( r x( r x a)) 2( r . a)<br />
<br />
iii) div{r<br />
n<br />
( a x r )} 0, n being<br />
a<br />
cons tan t.<br />
Suggested answer:<br />
<br />
Let a a1<br />
i a2<br />
j<br />
a3<br />
k<br />
Page 55 of 72
Engineering Mathematics - II - <strong>Vector</strong> Calculus - 2007<br />
<br />
r x i y j<br />
z k<br />
<br />
<br />
i j k<br />
i)<br />
a x r a1<br />
a2<br />
a3<br />
x y z<br />
<br />
<br />
<br />
( a2z<br />
a3y)<br />
i (a3x<br />
a1z)<br />
j<br />
(a1y<br />
a2x)<br />
k<br />
<br />
<br />
<br />
div( a x r ) (a2z<br />
a3y)<br />
(a3x<br />
a1z)<br />
(a1y<br />
a2x)<br />
0<br />
x<br />
y<br />
z<br />
<br />
ii) r x( r x a) ( r . a) r ( r . r ) a<br />
<br />
div( r x( r x a)) ( r . a). r ( r . a)div r div(r<br />
2 <br />
<br />
a)<br />
<br />
a . r ( r . a)3 a . r<br />
2<br />
<br />
r<br />
2<br />
div a<br />
<br />
a . r 3( a . r ) a{2r<br />
x<br />
r<br />
<br />
i <br />
y z <br />
2r j<br />
2r k}<br />
r r<br />
<br />
a . r 3( a . r ) 2( a . r )<br />
<br />
2( a . r )<br />
<br />
<br />
iii) div{r<br />
n<br />
( a x r )} r<br />
n<br />
.( a x r ) r<br />
n<br />
div( a x r )<br />
<br />
r <br />
nr<br />
n1<br />
.( a x r ) r<br />
n<br />
div( a x r )<br />
r<br />
Page 56 of 72
Engineering Mathematics - II - <strong>Vector</strong> Calculus - 2007<br />
<br />
0 r<br />
n<br />
div( a x r )<br />
<br />
<br />
i j k<br />
a x r a1<br />
a2<br />
a3<br />
x y z<br />
<br />
i(a2z a3y) j(a1z a3x) k(a1y a2x)<br />
<br />
<br />
<br />
div( a x r ) (a2z<br />
a3y)<br />
(a3x<br />
a1z)<br />
(a1y<br />
a2x)<br />
x<br />
y<br />
z<br />
= 0 + 0 + 0<br />
= 0<br />
div(r<br />
n <br />
( a x r )) 0<br />
27. If<br />
<br />
a <strong>and</strong><br />
<br />
b<br />
are constant vectors, prove that<br />
<br />
i) div{( a x r ) b} a . b<br />
<br />
ii) div { a x( r x b )} 2( a . b )<br />
<br />
iii) div {( a x r )x(b x r )} [a,<br />
b , r ]<br />
Suggested answer:<br />
<br />
Let a a1<br />
i a2<br />
j<br />
a3<br />
k<br />
<br />
b b1<br />
i b2<br />
j<br />
b3<br />
k<br />
Page 57 of 72
Engineering Mathematics - II - <strong>Vector</strong> Calculus - 2007<br />
<br />
c x i y j<br />
z k<br />
i)<br />
<br />
div[( a . r ) b ] ( a . r ). b a . r div b<br />
<br />
a . b ( a . r )(0)<br />
<br />
a . b<br />
<br />
ii) we have a x( r x b ) ( a . b ) r ( a . r ) b<br />
<br />
<br />
div{ a x( r x b )} div{( a . b ) r } div{( a . r ) b}<br />
<br />
( a . b )div r ( a x r ). b ( a . r )div b<br />
<br />
( a . b )3 a . b ( a r )0<br />
<br />
2( a . b )<br />
iii)Let<br />
<br />
a x r p<br />
<br />
( a x r )x(b x r ) p x(b x r )<br />
<br />
(p . r ) b (p . b ) r<br />
<br />
{( a x r ). r } b {( a x r ). b} r<br />
<br />
2( a . b )<br />
Page 58 of 72
Engineering Mathematics - II - <strong>Vector</strong> Calculus - 2007<br />
<br />
iii)Let a x r p<br />
<br />
( a x r )x(b x r ) p x(b x r )<br />
<br />
(p . r ) b (p . b ) r<br />
<br />
{( a x r ). r } b {( a x r ). b} r<br />
<br />
0 [ a, r , b ] r<br />
<br />
( a x r )x(b x r ) [a,<br />
<br />
b ,<br />
<br />
r ] r<br />
<br />
div{( a x r )x(b x r )} [a,<br />
b , r ]. r [a, b , r ]div r<br />
<br />
r . [a,<br />
b ,<br />
<br />
r ] [a, b , r ](3)<br />
...(1)<br />
<br />
a1<br />
a2<br />
a3<br />
[ a, b , r ] b1<br />
b2<br />
b3<br />
x y z<br />
a1(b<br />
2z<br />
b3y)<br />
a2(b1<br />
z b3x)<br />
a3(b1<br />
y b2x)<br />
<br />
<br />
<br />
<br />
[ a, b , r ] (a2b3<br />
a3b2<br />
) i ( a1b<br />
3 a3b1<br />
) j<br />
(a1b<br />
2 a2b1<br />
) k<br />
<br />
a x b<br />
<br />
Now r . [a,<br />
b , r ] r .( a x b )<br />
( a2b3<br />
a3b2<br />
)x (a3b1<br />
a1b<br />
3)y<br />
(a1b<br />
2 a2b1)z<br />
Page 59 of 72
Engineering Mathematics - II - <strong>Vector</strong> Calculus - 2007<br />
<br />
[a, b , r ]<br />
In (1)<br />
<br />
div{( a x r )x(b x r )} [a, b , r ] 3[a, b , r ]<br />
<br />
[a, b , r ]<br />
28.<br />
<br />
If a<br />
<strong>and</strong><br />
<br />
b<br />
are vectors, prove that<br />
<br />
1<br />
1 <br />
a . {b . } {3( a, r )(b . r ) r<br />
2<br />
<br />
( a . b )}.<br />
r r<br />
5<br />
Suggested answer:<br />
<br />
<br />
1<br />
1<br />
r<br />
b . <br />
b . <br />
r <br />
r<br />
2<br />
r<br />
<br />
1 <br />
(b . r )<br />
r<br />
3<br />
<br />
<br />
1<br />
b . <br />
<br />
<br />
<br />
r <br />
1 1<br />
(b . r ) (b . r ) <br />
<br />
3<br />
3<br />
r<br />
r <br />
<br />
<br />
1 3 r<br />
[b ] (b . r ). <br />
r<br />
3<br />
r<br />
4<br />
r<br />
<br />
<br />
1<br />
b . <br />
<br />
<br />
<br />
r <br />
<br />
b 3(b . r ) <br />
r<br />
r<br />
3<br />
r<br />
5<br />
<br />
<br />
<br />
<br />
1<br />
b 3(b . r )<br />
a . b . <br />
a . . a . r<br />
<br />
r <br />
r<br />
3 r<br />
5<br />
<br />
Page 60 of 72
Engineering Mathematics - II - <strong>Vector</strong> Calculus - 2007<br />
1<br />
<br />
r<br />
5<br />
<br />
<br />
<br />
3( a . r )(b . r ) ( a . b )r<br />
2<br />
<br />
<br />
<br />
<br />
<br />
<br />
29. If F (x y 1) i j<br />
(x y)k show that F .Curl F 0.<br />
Suggested answer:<br />
<br />
Curl F <br />
<br />
i<br />
<br />
x<br />
x y 1<br />
<br />
j<br />
<br />
y<br />
1<br />
<br />
k<br />
<br />
z<br />
(x y)<br />
<br />
i(<br />
1)<br />
j( 1)<br />
k( 1)<br />
<br />
Curl F i j<br />
k<br />
<br />
F .Curl F (x y 1)( 1)<br />
1(1) (x y)( 1)<br />
= -x - y - 1 + 1 + x + y<br />
= 0<br />
<br />
<br />
30. If F x<br />
2<br />
y i 2xz j<br />
2yz k find x(<br />
x<br />
F ).<br />
Suggested answer:<br />
<br />
x<br />
F <br />
<br />
i<br />
<br />
x<br />
x<br />
2<br />
y<br />
<br />
j<br />
<br />
y<br />
2xz<br />
<br />
k<br />
<br />
z<br />
2yz<br />
<br />
<br />
i(2z 2x) j(0) k( 2z<br />
x<br />
2<br />
)<br />
Page 61 of 72
Engineering Mathematics - II - <strong>Vector</strong> Calculus - 2007<br />
<br />
i j k<br />
<br />
x(<br />
x<br />
F ) <br />
x<br />
y<br />
z<br />
2z 2x 0 2z x<br />
2<br />
<br />
(2x 2) j<br />
31. Find the directional derivative of<br />
x<br />
2<br />
yz 4xz<br />
2<br />
at (1, -2, -1)<br />
<br />
in the direction 2 i j<br />
2 k .<br />
Suggested answer:<br />
<br />
<br />
<br />
(2xyz 4z<br />
2<br />
) i x<br />
2<br />
z j<br />
(x<br />
2<br />
y 8xz) k<br />
<br />
at<br />
(1,<br />
2,<br />
<br />
1)is 8 i j<br />
10 k<br />
Let<br />
<br />
c 2 i j<br />
2 k<br />
Directional derivative in the direction of<br />
<br />
c<br />
is<br />
<br />
(2 i j<br />
2 k)<br />
. c (8 i j<br />
10 k).<br />
3<br />
<br />
1<br />
(16<br />
3<br />
1 20)<br />
37<br />
<br />
3<br />
32. Find the directional derivative of 4xz<br />
3 3x<br />
2<br />
y<br />
2<br />
z at (2, -1, 2)<br />
<br />
in the direction of 2 i 3 j<br />
6 k .<br />
Page 62 of 72
Engineering Mathematics - II - <strong>Vector</strong> Calculus - 2007<br />
Suggested answer:<br />
<br />
<br />
<br />
<br />
(4x<br />
3<br />
6xy<br />
2<br />
z) i ( 6x<br />
2<br />
yz) j<br />
(12xz<br />
2<br />
3x<br />
2<br />
y<br />
2<br />
) k<br />
<br />
at<br />
(2, 1,2)<br />
<br />
8 i 48 j<br />
84 k<br />
<br />
c 2 i 3 j<br />
6 k<br />
1<br />
. c [8(2) 48( 3)<br />
(84)(6)]<br />
7<br />
376<br />
<br />
7<br />
33. Find the directional derivative of<br />
x<br />
2<br />
y<br />
2<br />
z<br />
2<br />
at (1, 1, -1)<br />
in the direction of the tangent to the curve x = e t , y = 1 + 2sint,<br />
z = t - cost, 1 t 1.<br />
Suggested answer:<br />
x<br />
2<br />
y<br />
2<br />
z<br />
2<br />
<br />
<br />
<br />
<br />
2xy<br />
2<br />
z<br />
2<br />
i 2x<br />
2<br />
yz<br />
2<br />
j<br />
2x<br />
2<br />
y<br />
2<br />
z k<br />
<br />
at<br />
(1, 1,<br />
<br />
1) 2 i 2 j<br />
2 k<br />
<br />
<br />
<br />
r e<br />
t<br />
i (1 2 sin t) j<br />
(t cos t) k<br />
<br />
d r<br />
dt<br />
at t<br />
0<br />
is<br />
<br />
i 2 j<br />
k<br />
Page 63 of 72
t <br />
<br />
d r<br />
dt<br />
<br />
d r<br />
dt<br />
<br />
Engineering Mathematics - II - <strong>Vector</strong> Calculus - 2007<br />
<br />
i 2 j<br />
k<br />
6<br />
directional derivative of in the direction of<br />
<br />
c<br />
is<br />
<br />
( i 2 j<br />
k)<br />
(2 i 2 j<br />
2 k).<br />
6<br />
<br />
1<br />
[2 4 2]<br />
6<br />
<br />
4<br />
6<br />
34. Find the equation of the tangent plane <strong>and</strong> the normal line to the surface z =<br />
x 2 + y 2 at (2, -1, 5).<br />
Suggested answer:<br />
x<br />
2<br />
y<br />
2<br />
z<br />
<br />
<br />
2x i 2y j<br />
k<br />
Equation to the normal line at (2, -1, 5) is<br />
x 2<br />
4<br />
<br />
y 1<br />
<br />
2<br />
z 5<br />
1<br />
Equation the tangent plane is 4(x - 2) + (-1)(y + 2)(-1)(z - 5) = 0<br />
i.e., 4x - 2y - z - 5 = 0<br />
Page 64 of 72
Engineering Mathematics - II - <strong>Vector</strong> Calculus - 2007<br />
35. Find the angle between the surface x 2 + y 2 + z 2 = 9 <strong>and</strong> z = x 2 + y 2 -3 at<br />
the point (2, -1, 2).<br />
Suggested answer:<br />
Let<br />
x<br />
2<br />
y<br />
2<br />
z<br />
2<br />
1 9<br />
x<br />
2<br />
y<br />
2<br />
2 z 3<br />
1<br />
<br />
2x i 2y j<br />
2z k<br />
1<br />
at<br />
(2, 1,2)<br />
<br />
4 i 2 j<br />
4 k<br />
2<br />
<br />
2x i 2y j<br />
k<br />
2<br />
<br />
at (2, 1,2) 4 i 2 j<br />
k<br />
Angle between the surfaces is the angle between 1 <strong>and</strong> 2.<br />
<br />
<br />
.<br />
cos<br />
1 1 2<br />
<br />
<br />
|<br />
1<br />
|| 2<br />
| <br />
<br />
<br />
4(4) ( 2)( 2) 4( 1)<br />
cos 1 <br />
<br />
36 21 <br />
<br />
<br />
8<br />
cos 1 <br />
3<br />
21 <br />
36. Find the values of a <strong>and</strong> b such that the surfaces ax 2 - byz = (a + 2)x <strong>and</strong><br />
4x 2 y + z 3 = 4 are orthogonal at (1, -1, 2).<br />
Suggested answer:<br />
Let : ax<br />
2<br />
1 byz (a 2)x 0<br />
...(1)<br />
Page 65 of 72
Engineering Mathematics - II - <strong>Vector</strong> Calculus - 2007<br />
: 4x<br />
2<br />
y z<br />
3<br />
2 4 0<br />
...(2)<br />
Given (1, -1, 2) is a point on (1) <strong>and</strong> (2)<br />
1 2<br />
at (1, 1,2)<br />
i.e., a + 2b - (a + 2) -4 + 8 - 4<br />
2b - 2 = 0<br />
b 1<br />
1<br />
<br />
(2ax (a 2)) i<br />
bz j<br />
by k<br />
<br />
1 at (1, 1,2) (a 2) i 2 j<br />
by k<br />
2<br />
<br />
8xy i 4x<br />
2<br />
j<br />
3z<br />
2<br />
k<br />
<br />
2<br />
at (1, 1,2) 8<br />
i 4 j<br />
12 k<br />
since 1 <strong>and</strong> 2<br />
are orthogonal<br />
1. 2<br />
0<br />
-8(a - 2) - 8 + 12 = 0<br />
i.e., a = 5/2<br />
a 5 / 2<br />
<strong>and</strong> b 1<br />
37. Find the constant a, b, c so that the vector<br />
<br />
<br />
<br />
<br />
f (x 2y az) i (bx 3y z) j<br />
(4x cy 2z)k<br />
is irrotational.<br />
Page 66 of 72
Engineering Mathematics - II - <strong>Vector</strong> Calculus - 2007<br />
Suggested answer:<br />
<br />
F is irrotational if<br />
<br />
Curl F 0<br />
i j k<br />
<br />
CurlF <br />
0<br />
x y z<br />
x 2y az bx 3y z 4x cy 2z<br />
c 1 0, 4 a 0, b 2 0<br />
a 4,b 2, c 1<br />
38. If <strong>and</strong> <br />
are scalar fields prove that<br />
i) div{ }<br />
<br />
2<br />
.<br />
<br />
ii) <br />
2<br />
( )<br />
<br />
2<br />
<br />
2<br />
2.<br />
<br />
Suggested answer:<br />
i) div( )<br />
.<br />
div( )<br />
. <br />
2 <br />
...(1)<br />
ii) similarly<br />
div( ) . <br />
2 <br />
...(2)<br />
adding (1) <strong>and</strong> (2)<br />
div( )<br />
div( )<br />
<br />
2<br />
<br />
2<br />
2.<br />
<br />
...(3)<br />
Also div( <br />
) div( ) div( ( )) <br />
2<br />
( )<br />
<br />
2<br />
( )<br />
<br />
2<br />
<br />
2<br />
2.<br />
<br />
39. Evaluate<br />
i)<br />
2 r<br />
2 <br />
Page 67 of 72
ii)<br />
Engineering Mathematics - II - <strong>Vector</strong> Calculus - 2007<br />
<br />
<br />
<br />
2 r<br />
div<br />
<br />
r<br />
2 <br />
<br />
Suggested answer:<br />
i) <br />
2<br />
r<br />
2<br />
<br />
<br />
2<br />
2<br />
2<br />
(r<br />
2 <br />
) (r<br />
2 <br />
) (r<br />
2<br />
)<br />
x<br />
2<br />
y<br />
2<br />
z<br />
2<br />
<br />
x y <br />
2r<br />
2r<br />
<br />
x<br />
r y<br />
r <br />
<br />
<br />
<br />
2r<br />
z <br />
z <br />
<br />
r <br />
= 2 + 2 + 2<br />
= 6<br />
ii)<br />
<br />
r 1 1<br />
div<br />
<br />
<br />
. r <br />
r<br />
2<br />
r<br />
2<br />
r<br />
2<br />
<br />
<br />
div r<br />
<br />
<br />
2 r 1<br />
.<br />
r (3)<br />
r<br />
3 r r<br />
2<br />
<br />
<br />
2 3 <br />
r<br />
2<br />
r<br />
2<br />
1<br />
<br />
r<br />
2<br />
40.<br />
3 Find the value of a if f(axy x) i(a 2)x j(1<br />
2 a )xz<br />
2<br />
<br />
k<br />
is irrotational.<br />
Suggested answer:<br />
Since<br />
<br />
f is irrotational<br />
Page 68 of 72
Engineering Mathematics - II - <strong>Vector</strong> Calculus - 2007<br />
<br />
Curl f 0<br />
<br />
i<br />
<br />
i.e.,<br />
x<br />
axy z<br />
3<br />
<br />
j<br />
<br />
y<br />
(a 2)x<br />
2<br />
<br />
k<br />
<br />
z<br />
(1 a)xz<br />
2<br />
0<br />
<br />
(1<br />
a)z<br />
2<br />
3z<br />
2<br />
k (a<br />
2)2x ax<br />
<br />
<br />
i{0 0} j<br />
0<br />
z 2<br />
1<br />
a 3 0 <strong>and</strong> x (a<br />
2)2 a 0<br />
a 4<br />
41. Show that<br />
<br />
r<br />
r<br />
3<br />
is both solenoidal <strong>and</strong> irrotational.<br />
Suggested answer:<br />
<br />
r 1<br />
div<br />
<br />
r<br />
3<br />
r<br />
3<br />
<br />
1 <br />
div r <br />
<br />
. r<br />
r<br />
3<br />
<br />
1<br />
(3) <br />
r<br />
3<br />
<br />
3 r <br />
.<br />
r<br />
r<br />
4 r <br />
<br />
3 3 <br />
r<br />
3<br />
r<br />
3<br />
= 0<br />
<br />
r 1 1 <br />
Curl<br />
Curl r <br />
<br />
x r<br />
r<br />
3 r<br />
3<br />
r<br />
3<br />
<br />
<br />
Page 69 of 72
Engineering Mathematics - II - <strong>Vector</strong> Calculus - 2007<br />
<br />
1 3 r <br />
(0) x<br />
r<br />
r<br />
3 r<br />
4 r <br />
<br />
<br />
0 0<br />
<br />
0<br />
42. If<br />
<br />
a is a constant vector prove the following<br />
i)<br />
<br />
<br />
<br />
<br />
Curl( a . r ) a 0<br />
<br />
<br />
ii)Curl{ r x( a x r )} 3( r x a)<br />
<br />
<br />
iii)Curl{r<br />
n<br />
( a x r )} (n 2)r<br />
n<br />
a nr<br />
n2<br />
( r . a) r<br />
Suggested answer:<br />
<br />
i)Curl{( a . r ) a} ( a . r )Curl a ( a . r )x a<br />
<br />
( a . r ) 0 a x a ( a . r ) a<br />
<br />
0 0<br />
<br />
0<br />
<br />
<br />
ii) Curl{ r x( a x r )} Curl{( r . r ) a ( r . a) r }<br />
<br />
<br />
Curl{( r . r ) a} Curl{( r . a) r }<br />
<br />
( r . r )Curl a ( r . r )x a ( r . a)x r<br />
Page 70 of 72
Engineering Mathematics - II - <strong>Vector</strong> Calculus - 2007<br />
<br />
r <br />
0 2r x a ( r . a) 0 a x r<br />
r<br />
<br />
2( r x a) r x a<br />
<br />
3( r x a)<br />
<br />
<br />
iii) Curl{r<br />
n<br />
( a x r )} r<br />
n<br />
Curl( a x r ) r<br />
n<br />
x( a x r )<br />
<br />
r <br />
r<br />
n<br />
Curl( a x r ) nr<br />
n1<br />
x( a x r )<br />
r<br />
<br />
<br />
r<br />
n<br />
Curl( a x r ) nr<br />
n2<br />
{( r . r ) a ( r . a) r }<br />
<br />
r<br />
n<br />
Curl( a x r ) nr<br />
n<br />
a nr<br />
n2<br />
( r . a) r<br />
...(1)<br />
<br />
<br />
i j k<br />
<br />
<br />
<br />
Consider ( a x r ) a1 a2<br />
a3<br />
i(a2z<br />
a3y)<br />
j(a3x<br />
a1z)<br />
k(a1y<br />
a2x)<br />
x y z<br />
<br />
Curl( a x r ) <br />
<br />
i<br />
<br />
x<br />
a2z<br />
a3y<br />
<br />
j<br />
<br />
y<br />
a3x<br />
a1z<br />
<br />
k<br />
<br />
z<br />
a1y<br />
a2x<br />
<br />
2 a<br />
<br />
in (1)<br />
<br />
Curl{r<br />
n<br />
( a x r )} 2r<br />
n<br />
a nr<br />
n<br />
a nr<br />
n2<br />
( r . a) r<br />
<br />
(n 2)r<br />
n<br />
a nr<br />
n2<br />
( r . a) r<br />
Page 71 of 72
Engineering Mathematics - II - <strong>Vector</strong> Calculus - 2007<br />
Page 72 of 72