- Fourier Series and Fourier Integrals An introduction, but at university level
- Differential geometry 1. Surfaces in 3-dimensional space. Students outline
Parameter curves. Differentiable surfaces. Shortest path, covariant derivative, Christoffel symbols and Gauss' theorem
- Differential geometry 2. Tensor analysis with application to General Relativity.
Students outline (pdf)
Linear algebra and tensors. Generalized coordinates i n-dimensions. Covariant derivative. Parallel transport. Riemanns curvature tensor.
- Taylors formula. (pdf)
A derivation of Taylor's formula with application to Maclaurin series of standard functions.
- Probability Theory. An introduction and beyond
A textbook on probability theory that goes beyond the introductory level.
- The Platonic solids. The five regular polyhedra
Proof of Eulers polyhedron theorem. The Dihedral angles, and radii of the inscribed and circumscribed spheres of the the five regular polyhedrons.
- Spherical geometry. A classical approach
The right angle spherical tringle. Cosine- and sine relations for the general spherical triangle. Area of a spherical triangle.
- Calculus of Variations. Applied to known and unknown problems
Euler-Lagrange equations. The simplest problem. Largest volume for a given surface. The suspended chain.
The Brachistochrone. On the shape of soap membranes. On the shape of wine barrels. On the shape of a hanging water drop
- Eigenvalue problems in linear algebra. (pdf)
A general discussion of algebra of matrices, and their eigenvalues. Illustrated by an example.
- Implicit_differentiation with examples. (pdf)
The equation for the tangent to an ellipse. The least bending of a beam through a prism in
- Games: Probabilities and strategies
Lotto, Poker, Casino. Ruin probabilities. Theory of strategies. The optimal strategy (Snell-strategy). Examples of using strategies.
- The birthday problem and other improbable probabilities
The coin in the three boxes, The card game "war",(number of permutations with no fixed elements), the Sct. Petersborg paradox
- The formula for the sums of n integers raised to a integer power
Derivation of a recursion formula for the sum of integer powers
- Treating the mathematics behind parallel and central projections
- Geometrical constructions of ovals and of the golden cut
Ovals in architecture and mathematical examples of the golden cut
- Achilles and the turtle
A mathematical explanation of a ancient Greek paradox.
- The number of bricks in a four sided pyramid.
The number of oranges in a three sided_pyramid
Deriving a solution to two classical problems.
- The Brachistocrone and the Tautocrone.
Two classical problems solved by advanced calculus
- Generalized Newton-Rapson method, and the method of steepest descent
Finding zero points of functions of several variables. Finding minimum of functions of several variables
- Vector Analysis
The gradient, divergence, and curl. Gauss' Stokes and Green theorems. Vector analysis in curvilinar coordinates.
- The peculiar Fibonacci numbers.
A note of the properties of the Fibonacci numbers.
- Elementary Group theory
Compositions in a set. Groups, rings and algebras. Formal algebraic extension of the real numbers to the complex numbers
- The complex number system
Calculation rules for complex numbers. The quadratic and the binome equation. de Moivre's and Eulers formulas. The fundamental theorem of algebra.
- Solution formula for the cubic (third degree) equation
Devivation of the complete solution to the third degree equation. (Cardanos solution)
- Analytic (holomorphic) functions
Demonstrating Cauchy's first and second integral theorem.
- Laurent series. Residues. Contour integrals
- The exact value of the sum of the reciprocals of n-square. The value of Zeta(2), Zeta(4)and Zeta(6)
- On queues in highways and before traffic lights.
- Economic models for small enterprises.
- Elementary national economics. A mathematical approach
- Submarine hunting and the logarithmic spiral
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