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

Engineering Mathematics - II - Vector Calculus - 2007
Let f f1
i f2
j f3
k
then f f1
i f2
j f3
k
div ( f ) ( f1 ) ( f2
) ( f3)
x y z
f1
x
f1
x
f
2
y
f2
y
f
3
y
f3
y
f1
x
i
x
. f1
i
div
f
.
f
Laplacian of a scalar field
Let
then
= (x,y,z) be a scalar field
is a vector field
and
div
( )
.
x
x
y
y
z
z
2
x
2
2
y
2
z
2
denoted by
2
is
called
the
Laplacian
of
a
scalar
field.
Note 1:
2
2
x
2
2
y
2
2
z
2
Note 2:
A
scalar
field
is
called
a harmonic
function if
2
0
Note 3:
2
()
2
, where
2
and are cons tan ts,
and
are scalar fields.
Curl of a Vector field
Let f f1
i f2
j f3
k
then
curl
of
f
denoted by
curl
f
or
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