[ Home | Lectures 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 ]

*Lecture 1
*[ Exercises ]

- Some shorthand notation
- Numbers:
**N, Z, Q, R**and**C**; polar representation - Complex arithmetic: addition, multiplication, inverse, complex conjugate
- Infinite sequences: independent/recursive representation, convergence, limit
- Infinite series: examples (geometric, harmonic, alternating harmonic), convergence

*Lecture 2
*[ Exercises ]

- Convergence tests: comparison test, Cauchy root test, D'Alambert or Cauchy ratio test, Cauchy or Maclaurin integral test*, Leibnitz criterion for alternating series
- Euler's number e
- Functions
- Representations: explicit, implicit, parametric
- Properties: zeros, maxima/minima; bounded from above/below, positive/negative definite, even/odd, x-y-symmetric, periodic, (strictly) monotonic
- Inverse
- Continuity: definition, examples for discontinuities (jump, gap, pole, oscillation)
- Limits: at a gap, at infinity

*Lecture 3
*[ Exercises ]

**Guest speaker Dr. Poethig: ***Computing at the Physics Department
and access to the Internet*

- Differentiation
- Definitions: differentiability, derivative, slope
- Higher derivatives
- Rules: sums and constant factors; product rule, quotient rule, chain rule; differentiation of inverse functions

*Lecture 4
*[ Exercises ]

- Examples: derivatives of polynomials,
rational functions,
*n-*th roots, trigonometric functions, inverse trigonometric functions, exponential functions, logarithms, hyperbolic and inverse hyperbolic functions - Extrema: finding minima and maxima of functions
- Partial derivatives
- Integration
- Definitions: indefinite/definite integral, fundamental theorem of calculus
- Rules: linearity, substitutions; rules for definite integrals
- Examples: basic integrals
- Methods: substitution; integration by parts; reducing the integrand to f'(x)/f(x)

*Lecture 5
*[ Exercises ]

- Taylor expansion
- Definitions: Taylor series, Maclaurin series
- Examples: binomial theorem, trigonometric functions, exponential function, logarithm
- Extending functions into the complex plane

*Lecture 6
*[ Exercises ]

- Euler-Moivre equation
- Fourier expansion
- Definition: Fourier series
- Why should a Fourier expansion exist?
- Coefficients: orthogonality relations, Fourier coefficients

*Lecture 7
*[ Exercises ]

- Examples: sawtooth wave, triangular wave, square wave
- Delta function
- Integral transforms
- Fourier transform
- Definition
- Properties: linearity, translations, symmetries, convolution ("Faltung") theorem, Fourier transform of derivatives, Parseval equation

*Lecture 8
*[ Exercises ]

- Laplace transform: definition, properties, examples

*Lecture 9
*[ Exercises ]

- Differential equations
- Definitions: ordinary/partial, order, explicit/implicit representation, system of coupled differential equations
- Solution: general/special, initial/boundary conditions
- Ordinary first order differential equations
- Separable variables
- Homogeneous equations
- Exact equations
- Integrating factors: application to linear first-order differential equations

*Lecture 10
*[ Exercises ]

- Ordinary linear differential equations
- Definitions: homogeneous/inhomogeneous, with constant coefficients, principle of superposition, linearly independent solutions

- Vector spaces
- Vectors: addition, multiplication by a real number, magnitude, unit vector
- Scalar product: definition, properties, Schwarz inequality, triangle inequality; orthogonality
- Linear independence and dimension
- Basis and components: orthonormal basis
- Vector product: definition, properties; Levi-Civita tensor; triple vector product
- Triple scalar product

- Matrices
- Linear maps and matrices: definitions
- Matrix operations: multiplication by a number, addition, matrix multiplication, transpose
- Special matrices: unit matrix, symmetric matrices, orthogonal matrices
- Inverse: properties, Gauss-Jordan algorithm
- Determinants: definition, properties; connection to triple scalar product

- Eigenvalues and eigenvectors: definitions; degenerate eigenvalues, eigenspace, spectrum; eigenvalues and -vectors of symmetric matrices; secular equation; kernel
- Diagonalization of a symmetric matrix: basis transformation; trace, determinant

- Forms
- Definitions: p-form, components
- Wedge product

- Functions, curves and tangent vectors
- Vector fields: definition; congruence of curves
- Directional derivative
- Coordinates: Cartesian, polar, general; coordinate lines
- Coordinate frames: local basis, transformation rule, Jacobian matrix

- Components of a vector field: definition, transformation rule
- Fields of p-forms: components, transformation rule; volume element
- Gradient
- Integration: integral of a volume element; substitution rule for multi-dimensional integrals
- Special coordinates in 3D