02647 ANVENDT MATEMATIK FOR FYSIKERE

Applied Mathematics for Physicists

Fall 2003

Textbook:

• G.B. Arfken and H.J. Weber,
  Mathematical Methods for Physicists,
  5'th Edition,
  Academic Press, San Diego (2001).
  

• Jon Mathews and R. L. Walker,
  Mathematical Methods of Physics,
  2nd Edition,
  Addison-Wesly Publishing Company, Redwood City, California (1970).

DAYS 1-2
Introduction and integral transformations

Lectures 1-3 (mps):

• Introduction and survey. Nonlinearity and linearization. Superposition.
• General integral transforms. Initial and boundary value problems.
• Laplace transformation. Dirac's delta-function.
• Read: pages 84-88, 905-907, 938-940, 942-944, and 946-948
  

Lectures 4-6 (ob):

• General integral transforms.
• Fourier transformation - different definitions.
• Fourier transformation - relation between the two spaces.
• Fourier transformation as solution technique .
• Convolution, Parceval, and transfer function.
• Read: pages 905-909, 911, 911-915, 920-922, 924-926, 935-936

DAYS 3-4
Sturm-Liouville problems and special functions from mathematical physics

Lectures 7-9 (mps):

• Sturm-Liouville Theory. Self-adjoint equations and operators.
• Eigenvalues and eigenfunctions. Quadratic forms.
• Eigenfunction expansion. Completeness of eigenfunction set. Asymptotics.
• MMP: sections from chapters 8-9 and 11-13.

Lectures 10-12 (mps):

• Reduction of partial differential equations by separation of variables.
• Solutions of ordinary differential equations.
• Special functions: Bessel functions, Legendre polynomials, and others.
• MMP: sections from chapters 8-9 and 11-13.

DAYS 5-6
Green's function

Lectures 13-15 (ob):

• Green's function as a tool for solving inhomogeneous problems.
• Expansion in eigen functions.
• Resonance and scattering problems. Existence of Greens function.
• MMP: chapters 8.7 and 9.5.

Lectures 16-18 (ob):

• Symmetry. Relation to self-adjoint operators.
• Systems of differential equations.
• MMP: chapters 8.7 and 9.5.

DAYS 7-8
Tensors

Lectures 19-21 (plc):

• Coordinate transformations
• MMP: Chapter 3.2-3.4.

Lectures 22-24 (plc):

• Tensor analysis.
• Pseudotensors.
• Non-Cartesian tensors.
• MMP: Chapter 2.6-2.11.

DAY 9
Group Theory

Lectures 25-27 (plc):

• Introduction to group theory.
• Continuous groups.
• Discrete groups.
• MMP: Selected sections from Chapter 4

DAYS 10-13
Calculus of variations

Lectures 28-30 (ob):

• Euler-Lagrange equation and applications.
• Generalizations. Several dependant and independant variables.
• MMP: chapter 17.

Lectures 31-33 (ob):

• Hamilton's principle.
• Lagrange multipliers. Variations subject to constraints.
• MMP: chapter 17.

Lectures 34-36 (ob):

• Rayleigh-Ritz variational technique.
• Overview.
• MMP: chapter 17.