§1.5-6 Review; Linear Function Spaces Christopher Crawford PHY

Outline Review for exam next class Chapter 1, Wednesday, October 1 Linear function spaces Basis – Delta function expansion Inner product – orthonormality and closure Linear operators – rotations and stretches, derivatives Inverse Laplacian and proof of Helmholtz theorem Particular solution of Poisson’s equation Proof of Helmholtz theorem 5 th annual Dr. Jekyll and Mr. Hyde contest 2

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 3

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 δ ij – Represents a continuous type of “orthonormality” of basis functions It is the kernel (matrix elements) of the identity matrix 4

Vectors vs. Functions 5

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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 8

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 chapter to recover Coulomb’s law from the Maxwell equations 9

5 th annual Dr Jekyll & Mr Hyde Contest Welcome 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, 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, st prize: 2% bonus credit (final grade), 2 nd prize: 1.5%, honorable mention: 1% (all other submissions). 10