University Notes

About These Notes

These notes cover rigorous, proof-based material at the undergraduate level. They are written in the style of a concise textbook: every definition is precise, every theorem is proved (or its proof is sketched with enough detail to complete), and every claim is justified from first principles.

The target audience is a second-year undergraduate STEM student who has completed first-year calculus and linear algebra, and is ready for the more abstract and demanding material that follows.

Scope

These notes are organised into three broad areas:

Mathematics

Topic	Description
Linear Algebra	Vector spaces, matrices, eigenvalues, inner product spaces
Real Analysis	Sequences, series, continuity, differentiability, integration
Multivariable Calculus	Partial derivatives, multiple integrals, vector calculus
Probability and Statistics	Probability spaces, distributions, estimation, hypothesis testing

Physics

Topic	Description
Classical Mechanics	Lagrangian and Hamiltonian formulations, Noether's theorem
Electromagnetism	Maxwell's equations, electrostatics, magnetostatics, waves
Quantum Mechanics	Postulates, Schrodinger equation, angular momentum, perturbation

Computing

Topic	Description
Algorithms and Data Structures	Complexity, sorting, graph algorithms, NP-completeness
Discrete Mathematics	Logic, proof techniques, combinatorics, graph theory, recurrences

How to Use These Notes

Each chapter is self-contained enough to serve as a reference, but the chapters within a subject are ordered by prerequisite dependencies. If you are working through linear algebra for the first time, read the chapters in order. If you need a quick reference for a specific result, the table of contents and section headings are designed for rapid navigation.

Mathematical content is rendered with KaTeX. Theorems, definitions, and proofs use Docusaurus admonitions where appropriate.

Notation Conventions

Symbol	Meaning
$\mathbb{R}, \mathbb{C}, \mathbb{Z}, \mathbb{N}, \mathbb{Q}$	Real, complex, integer, natural, rational numbers
$\forall, \exists$	Universal and existential quantifiers
$\implies, \iff$	Implies, if and only if
$\in, \subseteq$	Element of, subset of
$\mathbf{v}, \mathbf{A}$	Vectors and matrices (bold)
$\lVert \cdot \rVert$	Norm
$\langle \cdot, \cdot \rangle$	Inner product
$\nabla$	Gradient / del operator
$\partial$	Partial derivative / boundary

Contributing and Feedback

These notes are a living document. If you find an error or have a suggestion, open an issue or pull request on the GitHub repository.

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