PHYSICS 242   Advanced Calculus for Applications in Physics

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COURSE CONTENT

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• vector analysis
  • divergence and curl of vector fields
  • conservative vector fields
  • vector identities
  • integral calculus
    • line, surface and volume integrals of vector fields
    • Stoke's theorem
    • the divergence theorem
• curvilinear coordinates
  • plane polar coordinates
  • cylindrical coordinates
  • spherical polar coordinates
• Dirac Delta function
  • some properties satisfied by the Dirac delta function
  • point charges and the Dirac delta function,
    the Laplacian of the inverse-r potential
  • Dirac delta in curvilinear coordinates
• operations on integrals
  • some elementary techniques on evaluation of integrals:
    introducing complex varables, change of variable(s),
    differentiation or integration with respect to a parameter
  • integration by parts
  • Gauss numerical quadrature for integration,
    relevance to the classical orthogonal polynomials.
  • Leibnitz´s rule: differentiation of integrals involving a parameter
  • evaluation of some definite integrals using the residue theorem,
    contour integration in the complex plane
• gamma, beta and error functions
  • the gamma function: definite-integral representation, Stirling's asymptotic formula
    incomplete gamma functions and recursion relations
  • the Beta function, evaluation of some definite integrals in terms of the Beta function,
  • The error function: relationship to the normal probability density
• integral equations
  • classification: Fredholm and Volterra integral equations
    of homogeneous or inhomogeneous types and of the first or second kinds
  • transformation of a differential equation into an integral equation
  • converting a Volterra integral to an initial value problem
  • Neumann series solution
  • separable Kernel solution
  • approximations achieved by guessing the solution
  • solutions achieved by a suitable approximation for the Kernel
  • integral equations that involve generating functions of special polynomials
  • non-linear integral equations
  • integro-differential equations
• functions of a complex variable
  • Line integral of complex functions
  • analyticity, Cauchy-Riemann conditions
  • Stoke's theorem
  • Cauchy integral formula for a regular function f(z) and for its derivatives
  • singularities, poles, residues
  • residue theorem
  • contour integration: evaluation of definite integrals
    of rational functions of powers, exponentials, and trigonometric functions
  • treatment of functions involving branches
• Fourier series

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