1. §1.5 Delta Function; Function Spaces
Christopher Crawford
PHY 416

2. Outline
Example derivatives with singularities Electric field of a point charge – divergence singularity Magnetic field of a line current – curl singularity
Delta singularity δ(x) Motivation – Newton’s law: yank = mass x jerk Definition – differential of step function dϑ = δ dx Important integral identities Calculating with delta functions
Distributions – vs. functions Delta as an `undistribution’ Singularities and boundary conditions Building up higher dimensions: δ3(r)
Linear function spaces – functions as vectors Delta as a basis function or identity operator Correspondence table between vectors and functions

3. Example: magnetic field of a straight wire
This time: a singularity in the curl of magnetic intensity (flow)

4. Example: Inverse Square Law
Force of a constant carrier flux emanating from a point source

5. Newton’s law yank = mass x jerk force = mass x accel.
impulse = m x Δv singularities become more pronounced!

6. Delta singularity δ(x)
Differential definition: dϑ(x) = δ(x) dx Heaviside step function ϑ(x) = { 1 if x>0, 0 if x <0 }
Delta `function’ as a limit:

7. Important integral identities
Note the different orders of derivative
Offset delta function

8. Calculations with δ(x)
Jacobian
Higher dimension

9. Delta `undistribution’
Something you can integrate (a density)
The “distribution” of mass or charge in space
The delta `function’ is not well defined as a function
but it is perfectly meaningful as an integral
Think of δ(x) as an “undistribution”
The charge is clumped up into a singularity

10. Boundary conditions 2-d version of a PDE on the boundary
Derived from PDE by integrating across the boundary
RULES:
Proof:

11. δ(x) as a basis function
Each f(x) is a component for each x
Write function as linear combination
δ(x’) picks off component f(x)
The Dirac δ(x) is the continuous version of Kröneker δij
Represents a continuous type of “orthonormality” of basis functions
It is the kernel (matrix elements) of the identity matrix