§1.5 Delta Function; Function Spaces Christopher Crawford PHY 311 2014-01-31

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

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

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:

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

Calculations with δ(x) Jacobian Higher dimension

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

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

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

Vectors vs. Functions

Vectors vs. Functions

Vectors vs. Functions

Vectors vs. Functions