[ Home | Lectures 1
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Elementary Mathematical Methods for Physics
Lecture Notes
Basic concepts and terminology
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
Analysis
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 ]
Lecture 6
[ Exercises ]
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 ]
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
Lecture 10
[ Exercises ]
- Ordinary linear differential equations
- Definitions: homogeneous/inhomogeneous,
with constant coefficients, principle of superposition, linearly independent
solutions
Linear Algebra
Lecture 11
Lecture 12
Lecture 13
Lecture 14
Elementary Differential Geometry
Lecture 15