§1.5 Delta Function; Function Spaces

1. §1.5Delta Function;Function Spaces Christopher Crawford PHY 416 2014-09-24

2. Outline • Example derivatives with singularitiesElectric field of a point charge – divergence singularityMagnetic 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 identitiesCalculating with delta functions • Distributions – vs. functionsDelta as an `undistribution’Singularities and boundary conditionsBuilding up higher dimensions: δ3(r) • Linear function spaces – functions as vectorsDelta as a basis function or identity operatorCorrespondence 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 = massx accel. • impulse = m x Δvsingularities becomemore pronounced!

6. Delta singularity δ(x) • Differential definition: dϑ(x) = δ(x) dxHeaviside 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