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 />
<br />
Curl (Curl f ) grad (div f ) <br />
2<br />
f<br />
Proof:<br />
<br />
Let f f1<br />
i f2<br />
j f3<br />
k<br />
f3 f2 f1 f2 f2 f <br />
f i j<br />
1 <br />
k<br />
y z z1<br />
x x y <br />
<br />
Curl (Curl f )<br />
<br />
<br />
<br />
i <br />
<br />
f<br />
<br />
2<br />
y x<br />
<br />
<br />
<br />
f1<br />
<br />
<br />
y <br />
<br />
<br />
z<br />
<br />
<br />
<br />
f1<br />
z<br />
f <br />
<br />
3<br />
<br />
x <br />
<br />
<br />
2<br />
f <br />
2<br />
f <br />
2<br />
<br />
2<br />
<br />
<br />
<br />
1<br />
<br />
1 f<br />
<br />
1 f<br />
<br />
3<br />
i<br />
y<br />
x<br />
<br />
y<br />
2<br />
z<br />
2 z x <br />
<br />
<br />
<br />
2<br />
f <br />
2<br />
f <br />
2<br />
f <br />
2<br />
<br />
<br />
3<br />
<br />
2<br />
<br />
2 f<br />
<br />
1<br />
j<br />
z<br />
y<br />
<br />
z<br />
2<br />
x<br />
2 x y <br />
<br />
<br />
<br />
2<br />
<br />
2<br />
f <br />
2<br />
f f <br />
2<br />
1 3 3 f2<br />
<br />
k<br />
x<br />
z<br />
<br />
x<br />
2<br />
y<br />
2 y z <br />
<br />
adding<br />
<strong>and</strong><br />
subtracting<br />
<br />
2<br />
f1<br />
,<br />
y<br />
2<br />
<br />
2<br />
f3<br />
z<br />
2<br />
to<br />
each<br />
of<br />
the<br />
terms.<br />
<br />
<br />
<br />
<br />
i <br />
<br />
<br />
<br />
x<br />
f1<br />
<br />
x<br />
f<br />
<br />
2<br />
y<br />
f<br />
<br />
<br />
2<br />
<br />
<br />
3 f<br />
<br />
1<br />
z<br />
<br />
<br />
x<br />
2<br />
<br />
2<br />
f<br />
<br />
2<br />
y<br />
2<br />
<br />
2<br />
f <br />
<br />
<br />
1<br />
<br />
<br />
z<br />
2<br />
<br />
<br />
<br />
<br />
i (div f )<br />
x<br />
<br />
<br />
<br />
2<br />
f<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 />
<br />
<br />
<br />
<br />
f . <br />
f1<br />
f2<br />
f<br />
<br />
3<br />
<br />
x y<br />
<br />
z<br />
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