Mathematical Tools II - Advanced Multivariate Calculus

Abstract

In this course we will give a brief review of more dimensional calculus. The main focus of the course is vector analysis (integral theorems) and PDEs.

Objective

Students understand mathematics as a language for modelling and as a tool for solving practical engineering problems. They can analyse models, describe solutions qualitatively or calculate them explicitly if need be. They can solve examples as well as their practical applications manually and using computer algebra systems.

Content

Week 1 Day 1 – Revision more dimensional calculus • Discussion of self assessment. • More dimensional differentiation (partial derivatives, directional derivatives, gradient, extrema etc.). • More dimensional integration (iterated integrals, Fubini, change of coords (polar, cylindrical, spherical), Jacobi determinant) • Physical applications Prerequisites: • 1-dimensional differentiation and integration Day 2 – Vector analysis 1: Vector fields • Revision Vector fields • Line integrals • 2D flux and circulation • Fundamental theorem for line integrals Prerequisites: • Some knowledge of vector fields • 1-dimensional integration Day 3 – Vector analysis 2: • Green’s Theorem (2D versions of Stokes and Gauss) • Surface integrals Prerequisites: • vector product (interpretation of vector and of its length) Day 4 – Vector analysis 3: • Divergence and Rotation • Gauss’s Theorem (or divergence Theorem) • Stokes’s Theorem • Applications Day 5 – Revision ODEs 1st and 2nd order and Laplace transforms • Odes 1st and 2nd order • Laplace transforms • Heaviside- and δ-function • Summary of Week 1 Prerequisites: • Mathematical Tools I • Methods for solving ODEs 1st order (separation of variables, variation of constant) Week 2 Day 6 – Introduction and classification of PDEs: • General introduction • Classification • Terminology (Dirichlet, Neumann, mixed problems) Day 7 – Wave equation (1D and 2D) • Separating Variables • (double) Fourier Series • d'Alembert’s solution • method of characteristics • Steady State solution Day 8 – Heat equation • Fourier series • Fourier integrals • Fourier transforms Prerequisites: • Fourier series Day 9 – Laplace equation • Polar coordinates -> Fourier-Bessel series • cylindrical and spherical coordinates -> Euler-Cauchy • Using Laplace transforms Prerequisites: • Change of coordinates Day 10 – Reserve time • Summary • Preparation for the exam

Resources