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 />
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
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