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 />
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 />
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