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 />
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 />
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