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§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
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Example: magnetic field of a straight wire
• This time: a singularity in the curl of magnetic intensity (flow)
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Example: Inverse Square Law
• Force of a constant carrier flux emanating from a point source
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Newton’s law
• yank = mass x jerk
• force = mass x accel.
• impulse = m x Δv
singularities become
more pronounced!
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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:
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Important integral identities
• Note the different orders of derivative
• Offset delta function
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Calculations with δ(x)
• Jacobian
• Higher dimension
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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
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Boundary conditions
• 2-d version of a PDE on the boundary
• Derived from PDE by integrating across the boundary
• RULES:
• Proof:
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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
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