Syllabus Vector Differentiation - Velocity and Acceleration - Gradient ...
Syllabus Vector Differentiation - Velocity and Acceleration - Gradient ...
Syllabus Vector Differentiation - Velocity and Acceleration - Gradient ...
Create successful ePaper yourself
Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.
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
Change language