§1.5 Delta Function; Function Spaces Christopher Crawford PHY 416 2014-09-24

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

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

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

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

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

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