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1.5-6 Review Linear Function Spaces

• Christopher Crawford
• PHY 416
• 2014-09-29

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Outline

• Review for exam next classChapter 1, Wednesday,
  October 1
• Linear function spacesBasis Delta function
  expansionInner product orthonormality and
  closureLinear operators rotations and
  stretches, derivatives
• Inverse Laplacian and proof of Helmholtz
  theoremParticular solution of Poissons
  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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d(x) as a basis function

• Each f(x) is a component for each x
• Write function as linear combination
• d(x) picks off component f(x)
• The Dirac d(x) is the continuous version of
  Kröneker dij
• 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 Poissons 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 intoa) longitudinal
  and b) transverse components, which
  derive from a) scalar and b) vector
  potentials
• Proof project and invert the Lapacian, solve
  with Greens fns.
• Note we use the Helmholtz theorem next
  chapterto recover Coulombs law from the Maxwell
  equations

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5th 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, 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).