Mathematical Topics in Physics

This page will serve as an archive of the following topics in mathematical physics:

I. Tensors: Textbook: Neuenschwander’s Tensor Calculus for Physics

A. Vectors

B. Covectors (Dual Vectors)

C. Tensor Products

D. The Metric Tensor

E. Tensor Algebra

F. Tensor Calculus

II. Group Theory: Textbooks: Gallian’s Contemporary Abstract Algebra & Zee’s Group Theory in a Nutshell for Physicists.

A. Basic Definitions and Examples

Introduction to Groups
Properties of Groups

B. Subgroups and Permutation Groups
Definitions and Examples 
Properies of Subgroups and Permutation Groups

C. Isomorphisms & Homomorphisms
Definitions and Examples 
Properties of Isomorphisms and Homomorphisms

III. Real Analysis: Stephen Abbott’s Understanding Analysis and Maxwell Rosenlicht’s Introduction to Analysis:

Point-Set Topology

A. Metric Spaces
Introduction to Metric Spaces

B. Completeness
CONVERGENT SEQUENCES, CAUCHY SEQUENCES, COMPLETENESS
Basic Theorems of Completeness

C. Compactness
Open covers, Finite Subcovers, and COMPACTNESS
Basic Theorems of Compactness 

D. Connectedness
Definitions and Examples 
Basic Theorems of Connectedness

Calculus I and II Revisited

E. Continuous Functions

F. Differentiation

G. Integration

IV. Toplogy: Munkres’ Topology
Topics to be determined. 

The objective of this page is to serve as a reference for the pure mathematics that is used in physics. My studies recently have me looking at the mathematical side of physics, and so the posts here will reflect my current coursework. Each of the topics will have their own post, but this list is subject to revision and/or addition. As I encounter new topics I will add them to the list in my own time.


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