§1.5-6 Review; Linear Function Spaces

Christopher CrawfordPHY 416

2014-09-29

Outline• Review for exam next class

Chapter 1, Wednesday, October 1• Linear function spaces

Basis – Delta function expansionInner product – orthonormality and closureLinear operators – rotations and stretches, derivatives

• Inverse Laplacian and proof of Helmholtz theoremParticular solution of Poisson’s equationProof of Helmholtz theorem5th annual Dr. Jekyll and Mr. Hyde contest

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Review for exam• Linear Space – Vectors

– Basis, components; dot, cross, triple products; operators– Transformations: change of basis, coordinate transformations

• Differential Space – Derivatives and Integrals– Calculate the Gradient, Curl, Divergence, Laplacian– Calculate Line, Surface, Volume integrals

• Fundamental Theorems – Linear/differential structure– Apply the Gradient, Stokes’, Gauss’ theorems; integration by parts– Calculate with Delta functions; prove Helmholtz theorem– Essay question: geometrical interpretation of fields: Flux / Flow

• You are allowed one double-sided 8½x11 formula sheet– JUST formulas! NO pictures, descriptions, solved problems, examples

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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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Vectors vs. Functions

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Vectors vs. Functions

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Vectors vs. Functions

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General solution to Poisson’s equation• Expand f(x) as linear combination of delta functions• Invert linear Lapacian on each delta function individually

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Proof of the Helmholtz theorem• Theorem: Any vector field can be decomposed into

a) longitudinal and b) transverse components, which derive from a) scalar and b) vector potentials

• Proof: project and invert the Lapacian, solve with Green’s fns.

• Note: we use the Helmholtz theorem next chapterto recover Coulomb’s law from the Maxwell equations

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5th annual Dr Jekyll & Mr Hyde ContestWelcome to the fourth annual "Dr. Jekyll and Mr. Hyde" contest. You are each invited to submit a short (1-3 paragraphs) answer to the question: Which one of the electric flux (field lines) or electric flow (equipotentials) is more like Dr. Jekyll and which is more like Mr. Hyde? Why?I will post all submissions to the course website on Friday, 2014-10-03 before class.Your submissions will be "peer-reviewed" by yourselves, under criteria: physical insight, persuasiveness, cleverness, and humour. Each student may cast one secret vote by Doodle poll.The winner will be announced inclass on Monday, 2014-10-06. 1st prize: 2% bonus credit (final grade), 2nd prize: 1.5%, honorable mention: 1% (all other submissions).

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