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 />
i (div f ) <br />
2<br />
f<br />
x<br />
Note 1:<br />
<br />
(div f )<br />
<br />
<br />
<br />
2<br />
f<br />
<br />
<br />
The operator f . (f i f j f k) . <br />
1 2 3 <br />
<br />
<br />
i <br />
x<br />
<br />
<br />
<br />
f1<br />
f2<br />
f3<br />
x<br />
y<br />
z<br />
<br />
j<br />
y<br />
<br />
<br />
<br />
k <br />
z <br />
<br />
is called the Linear differential operator.<br />
Note 2: If is a scalar field<br />
Note 3:<br />
If<br />
<br />
<br />
<br />
<br />
f . <br />
f1<br />
f2<br />
f<br />
<br />
3<br />
<br />
x y<br />
<br />
g is a vector field<br />
<br />
z<br />
<br />
g g<br />
f . g f1<br />
f2<br />
f3<br />
x<br />
y<br />
<br />
g<br />
z<br />
(iii)<br />
<strong>Gradient</strong> of a scalar field:<br />
<br />
If f<br />
<br />
<strong>and</strong> g are vector fields<br />
then<br />
<br />
<br />
( f . g)<br />
<br />
<br />
f<br />
<br />
Curl<br />
<br />
g<br />
<br />
<br />
g<br />
<br />
Curl<br />
<br />
f<br />
<br />
<br />
( f . )<br />
g<br />
<br />
<br />
( g . )<br />
f<br />
Proof:<br />
Consider<br />
<br />
( f . g) i ( f . g)<br />
x<br />
<br />
<br />
<br />
<br />
<br />
f g <br />
i . g f . <br />
x x <br />
<br />
<br />
<br />
<br />
<br />
<br />
f <br />
i g . <br />
x <br />
<br />
<br />
<br />
<br />
g <br />
i f . <br />
x <br />
<br />
...(1)<br />
<br />
Now g Curl<br />
<br />
f<br />
<br />
<br />
g <br />
<br />
<br />
i<br />
<br />
<br />
f<br />
x<br />
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