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

## Elementary Mathematical Methods for Physics Lecture Notes

### Basic concepts and terminology

Lecture 1
[ Exercises ]

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 ]

Lecture 10
[ Exercises ]

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

Lecture 11

Lecture 12

Lecture 13

Lecture 14

Lecture 15