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 />
<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
Change language