Chapter 1. Vector Analysis
1.5 The Dirac Delta Function
1.5.1 The Divergence of
Consider the vector function directed radially:
Let’s apply the divergence theorem to this function:
Does this mean that the divergence theorem is false? What's going on here?
The divergence theorem MUST BE right since it’s a fundamental theorem.
The source of the problem is the point r = 0, where v blows up!
( ) vanishes everywhere except r = 0, its integral must be 4.
The entire contribution of must be coming from the point r = 0!
No ordinary function behaves like that.
It's zero except at the source location, yet its integral is finite!
It’s called the Dirac delta function.
It is, in fact, central to the whole theory of electrodynamics.
1.5.2 The One-Dimensional Dirac Delta Function
infinitely high,
infinitesimally narrow "spike,"
If f(x) is some "ordinary" function,
Under integral, picks out the value of f(x) at x = a.
Example 1.14
pick out the value of x3 at the point x = 2 23 = 8. Would be zero!
Example 1.15
Show thatZero everywhere except at x = a.
1.5.3 The Three-Dimensional Dirac Delta Function
Generalize the delta function to three dimensions:
with its volume integral is 1:
As in the one-dimensional case, integration with picks out the value f at r = 0.
The divergence of is zero everywhere except at the origin.
The integral of over any volume containing the origin is a constant (= 4)
More generally,
Since
or
The Three-Dimensional Dirac Delta Function:
Example 1.16 Evaluate the integral
Solution 1: It demonstrates something of the power and beauty of the delta function.
Solution 2: It is much more cumbersome but serves to illustrate the method of integration by parts.
Using the relation of partial integral:
on the boundary (where r = R),
1.6 The Theory of Vector Fields
1.6.1 The Helmholtz Theorem
Maxwell reduced the entire theory of electrodynamics to four differential equations, specifying respectively the divergence and the curl of E and B.
Since E and B are vectors, the differential equations naturally involve vector derivatives: divergence and curl.
Maxwell's formulation raises an important mathematical question:
To what extent is a vector function determined by its divergence and curl?
Or, can we determine the function F if its divergence and curl are specified?
To solve a differential equation appropriate boundary conditions are required.
In electrodynamics we typically require that the fields go to zero "at infinity" (far away from all charges).
The Helmholtz theorem guarantees that
the field, E or B is uniquely determined by its divergence and curl.
(Appendix B: A proof of the Helmholtz theorem)
Appendix B: A proof of the Helmholtz theorem
Suppose that the divergence and curl of a vector function F(r) are specified by
where C(r) must be divergenceless, because the divergence of a curl is always zero.
Question: can we, on the basis of this information, determine the function F?
If D(r) and C(r) go to zero sufficiently rapidly at infinity, the answer is yes!
“Helmholtz theorem”
(Proof) Assume that where
Then, (the divergence of a curl is zero at W)
(the curl of a gradient is zero at U)
'
1 1 1
( ) '( ) ( ) since
r r
0
Helmholtz
theorem
Corollary
Appendix B: The Helmholtz theorem
Helmholtz theorem
Corollary
where
For example, in electrostatics
In magnetostatics,
(V: Scalar potential)
(A: Vector potential)
1.6.2 Potentials
If the curl of a vector field (F) vanishes (everywhere),
then F can be written as the gradient of a scalar potential (V) Note the two null identities
the curl of the gradient of any scalar field is identically zero:
The divergence of the curl of any vector field is identically zero:
(The minas sign is purely conventional.)
Theorem 1: Curl-less fields (or "irrotational“ fields)
If the divergence of a vector field (F) vanishes (everywhere), then F can be written as the curl of a vector potential (A)
V 0
0
A
0
F
F V
0
F
F A Theorem 2: Divergence-less fields (or “solenoidal“ fields)
For all cases, any vector field can be written as F V A
0 F
F A
0 F
F V
