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PHY481 - Lecture 3
Sections 2.2-2.5 of Pollack and Stump (PS)
The Divergence
The divergence is the dot product of the gradient operator and a vector function,
F =
(Fx, Fy, Fz ), so that
F =
∂F
x
∂x
∂F
y
∂y
∂F
z
∂z
We want to find a co-ordinate independent representation of the divergence, which we achieve
by considering a small cube of dimension ǫ. Now consider an integral of the flux through
this surface, that is,
∮
dS
F · ndAˆ =
3 ∑
i=
[F
i
(~x + ǫˆe i
/2) − F
i
(~x − ǫˆe i
/2)]ǫ
2
(2)
where (ˆe 1
, ˆe 2
, eˆ 3
) = (ˆi,
j,
k) are introduced to simplify the expression. Eq. (22) reduces to,
∮
dS
F · ndAˆ =
3 ∑
i=
∂F
i
(~x)
∂x i
ǫ
3
= (
F )ǫ
3
(3)
The co-ordinate independent representation of the divergence of a vector function is then,
F = limV → 0
V
∮
dS
F · d
A (4)
The divergence is then proportional to the flux of the function,
F though the surface of the
volume V.
The Laplacian
The Laplacian ∇
2 is a scalar operator found by taking the dot product of the gradient
operator with itself, ie.,
2
=
∇; in cartesion co − ordinates ∇
2
=
2
∂x
2
2
∂y
2
2
∂z
2
The Curl
The curl of a vector function,
F is defined in the same way as the cross product
F =
∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣
i
j
k
x
y
z
F
x
F
y
F
z
∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣
∂F
z
∂y
∂F
y
∂z
∂F
x
∂z
∂F
z
∂x
∂F
y
∂x
∂F
x
∂y
Using Levi-Civita and suffix notation, this is,
F )
i
= ε ijk
∂F
k
∂x j
To find a co-ordinate independent representation, consider a square loop placed in the x-y
plane, with edge length ǫ. Consider a path integral around the loop, the circulation,
∮
loop
F · d
l = ǫF i
(~x − ǫeˆ j
/2) + ǫF j
(~x + ǫˆe i
/2) − ǫF i
(~x + ǫˆe j
/2) − ǫF j
(~x − ǫˆe i
This reduces to,
∮
loop
F · d
l = (
∂F
j
∂xi
∂F
i
∂xj
F )
k
ǫ
2
(9)
The general co-ordinate independent form of
F is then
nˆ · (
F ) = lim A→ 0
A
∮
C
F · d
l (10)
F. Derivative operator identities
Table 2.2 of PS gives a list of indentities for derivative operators. We will look at a couple
of interesting examples and in the homework you will need to use both these identities and
the vector identities in Table 2.1 of the text.
Identity.
∇f ) = 0. Proof: (
∇f ) i
= ε ijk
∂
2 f
∂x j
∂x k
. Because there is a sum over j, k
pairs of terms appear with opposite signs, due to the assymmetry of the Levi-Civita tensor.
Summing these pairs of terms gives zero.
Identity.
F ) = 0. Proof: Using Levi-Civita notation, the LHS is ε ijk
∂
2 F k
∂xi∂xj
Again the terms on the RHS can be collected into pairs with opposite signs due to the
antisymmetric nature of the Levi-Civita tensor. The identity is then proved.
G. Summary of key geometrical concepts
This has been a busy lecture with many mathematical details. If you have not seen these
mathematical tools before, it will take some time for you to become familiar with them.
However the most important lessons to take from these tools is the key concepts that enable
insight into the physics behind the math. Here are four key concepts.
- The cross product
A ∧
B is perpendicular to both
A and
B.
- The definition of the gradient through df = d~x ·
∇f demonstrates that
∇f is perpen-
diclar to equipotentials of the function f.
