Syllabus Vector Differentiation - Velocity and Acceleration - Gradient ...

Engineering Mathematics - II - Vector Calculus - 2007
i . ( f g)
x
i
.
f
x
g
f
g
x
f
i .
x
g
g
i .
x
f
i
f
.
x
g
i
g
.
x
f
( f ) .
g
(
g) .
f
g . Curl
f
f . Curl
g
(ii)
Curl of Curl of a vector:
Curl (Curl f )
grad (div
f )
2
f
Proof:
Let f f1
i f2
j f3
k
f3 f2 f1 f2 f2 f
f i j
1
k
y z z1
x x y
Curl (Curl f )
i
f
2
y x
f1
y
z
f1
z
f
3
x
2
f
2
f
2
2
1
1 f
1 f
3
i
y
x
y
2
z
2 z x
2
f
2
f
2
f
2
3
2
2 f
1
j
z
y
z
2
x
2 x y
2
2
f
2
f f
2
1 3 3 f2
k
x
z
x
2
y
2 y z
adding
and
subtracting
2
f1
,
y
2
2
f3
z
2
to
each
of
the
terms.
i
x
f1
x
f
2
y
f
2
3 f
1
z
x
2
2
f
2
y
2
2
f
1
z
2
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