Example
:
Find the curl for the function
2
ˆ

=
ρ
k
ρ
F
r
, where
k
is a constant and
ρ
is the
magnitude of the position vector in
xy
plane.
1.5.2. Second order spatial derivatives
1.5.2.1.
The Laplacian
The Laplacian of a scalar
f
is
∂
∂
∂
∂
+
∂
∂
∂
∂
+
∂
∂
∂
∂
=
∇
⋅
∇
≡
∇
3
3
2
1
3
2
2
1
3
2
1
1
3
2
1
3
2
1
2
1
q
f
h
h
h
q
q
f
h
h
h
q
q
f
h
h
h
q
h
h
h
f
f
(1.42)
In cylindrical coordinates it becomes
2
2
2
2
2
2
1
1
1
1
1
1
1
z
f
f
f
z
f
z
f
f
f
∂
∂
+
∂
∂
+
∂
∂
∂
∂
=
∂
∂
∂
∂
+
∂
∂
∂
∂
+
∂
∂
∂
∂
=
∇
φ
ρ
ρ
ρ
ρ
ρ
ρ
φ
ρ
φ
ρ
ρ
ρ
ρ
(1.43)
In these two equations, replacing scalar
f
by vector
F
r
results in the Laplacian of vector
F
r
.
Exercise:
Express equation (1.42) in (i) rectangular and (ii) spherical coordinates.
1.5.2.2.
The divergence of the curl of a vector
From equations (1.38) and (1.40), one gets
12
(
)
(
)
(
)
(
)
(
)
(
)
(
)
∂
∂

∂
∂
∂
∂
+
∂
∂

∂
∂
∂
∂
+
∂
∂

∂
∂
∂
∂
=
×
∇
⋅
∇
2
1
1
1
2
2
3
3
2
1
1
3
3
3
1
1
2
3
2
1
3
2
2
2
3
3
1
3
2
1
1
1
1
q
h
F
q
h
F
q
h
h
h
q
h
F
q
h
F
q
h
h
h
q
h
F
q
h
F
q
h
h
h
F
r
(1.44)
Exercise:
Express equation (1.44) in (i) Cartesian, (ii) cylindrical and (ii) spherical coordinates.
1.6.
Integral calculus
1.6.1.
Line, surface and volume integrals
The most important integrals in electromagnetism are the line (path) integral, surface
integral (or flux) and volume integral.
1.6.1.1.
Line integral
The line integral of a vector function along a prescribed path between points
a
and
b
is
∫
⋅
b
a
l
r
r
d
F
where we take the dot product of the vector
F
r
and the elemental length
l
r
d
at each point on the
path, with
l
r
d
being tangential to the path. If the path is closed then the integral is written as
∫
⋅
cl
d
l
r
r
F
In general these integrals depend on the path taken. There is a special class of vectors for which
the line integral does not depend on the path connecting the two points
a
and
b
. A force having
this property is known as a
conservative
quantity, and its line integral, which is the work done by
it, is independent of he path taken.
Example 1
:
Find the line integral of a vector
2
ˆ

=
cr
r
v
r
, with
c
a constant, from
r
=
r
1
to
r
=
r
2
.
Soln
.: In spherical coordinates the displacement is
φ
θ
d
ˆ
d
ˆ
d
ˆ
d
φ
θ
r
+
+
=
r
r
l
r
. Then

=
=
⋅
+
⋅
+
⋅
=
⋅
∫
∫
∫
∫
∫




2
1
2
2
2
2
1
1
d
d
sin
ˆ
ˆ
d
ˆ
ˆ
d
ˆ
ˆ
d
2
1
2
1
2
1
2
1
2
r
r
c
r
cr
r
cr
r
cr
r
cr
r
r
r
r
φ
φ
θ
θ
φ
θ
θ
φ
r
θ
r
r
r
v
l
l
1
l
r
r
where the rules for dot products of unit vectors have been applied.