1. §1.5 Delta Function; Function Spaces
Christopher Crawford
PHY 311

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

4. 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:

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

6. Calculations with δ(x)
Jacobian
Higher dimension

7. Distribution 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

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

9. 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

10. Vectors vs. Functions

11. Vectors vs. Functions

12. Vectors vs. Functions

13. Vectors vs. Functions