M.Sc - Higher Mathematics I
Welcome to the code resource and practice materials for the lecture Higher Mathematics I where our focus will be on the topic of ordinary differential equations and vector calculus.
All code presented were written in SageMath and have been tested on macOS Sonoma 14.5.
SageMath is a computer algebra system (CAS) with features covering many aspects of mathematics, including algebra, combinatorics, graph theory, group theory, differentiable manifolds, numerical analysis, number theory, calculus and statistics [Link].
The versions of the software is as follows
sage --version && python --version
SageMath version 10.3, Release Date: 2024-03-19 Python 3.12.5
There can be a wide variety of packages and dependencies required for the code snippets to work, therefore don't skip on a code while working through the examples.
| ORDER | TOPIC | DESCRIPTION |
|---|---|---|
| 1 | Introduction | Discussion of the lecture structure and what will be covered |
| 2 | First-Order ODEs | Separable ODEs, Exact ODEs, Linear ODEs |
| 3 | Second-Order Linear ODEs | Homogeneous Linear ODEs, Euler-Cauchy Equations |
| 4 | Higher-Order Linear ODEs | Homogeneous and non-homogeneous Linear ODEs |
| 5 | Systems of ODEs. Phase Plane. Qualitative Methods | Constant-Coefficient Systems. Phase Plane Method |
| 6 | Series Solutions of ODEs & Special Functions | Legendre’s Equation. Legendre Polynomials |
| 7 | Laplace Transforms | Laplace Transform. Linearity. First Shifting Theorem |
| 8 | Linear Algebra I: Matrices and Vectors | Linear Systems of Equations, Gauss Elimination |
| 9 | Linear Algebra II: Eigenvalue Problems | Determining Eigenvalues and Eigenvectors |
| 10 | Vector Calculus I: Grad, Div, Curl | Vector and Scalar Functions and Their Fields |
| 11 | Vector Calculus II: Integral Theorems | Path Independence of Line Integrals |
| 12 | Appendix | Tangental information related to the main topic |
Please read the code and all relevant information carefully as the wrong order of execution may cause issues.