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 />
<br />
( f x g) (y<br />
3<br />
z<br />
3<br />
) (z<br />
3<br />
x<br />
3<br />
) (x<br />
3<br />
y<br />
3<br />
)<br />
0<br />
<br />
f x g is solenoidal.<br />
25. Prove that<br />
<br />
div (r<br />
n<br />
r ) (n 3)r<br />
n<br />
.<br />
Suggested answer:<br />
<br />
div (r<br />
n<br />
r ) r<br />
n<br />
. r r<br />
n<br />
div r<br />
<br />
nr<br />
n1<br />
<br />
x<br />
r<br />
<br />
i nr<br />
n1<br />
y<br />
r<br />
<br />
<br />
<br />
<br />
z<br />
nr<br />
n 1<br />
k.<br />
r r<br />
n<br />
<br />
r <br />
<br />
<br />
x <br />
x<br />
<br />
<br />
y <br />
y<br />
<br />
<br />
z<br />
<br />
nr<br />
n1<br />
<br />
{x i y j<br />
z k}. r r<br />
n<br />
.(3)<br />
r<br />
nr<br />
n2<br />
.r<br />
2<br />
r<br />
n<br />
(3)<br />
( n 3)r<br />
n<br />
26. If<br />
<br />
a is a constant vector then prove that<br />
<br />
i) div( a x r ) 0<br />
<br />
ii) div ( r x( r x a)) 2( r . a)<br />
<br />
iii) div{r<br />
n<br />
( a x r )} 0, n being<br />
a<br />
cons tan t.<br />
Suggested answer:<br />
<br />
Let a a1<br />
i a2<br />
j<br />
a3<br />
k<br />
Page 55 of 72
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