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Lectures on Theoretical Physics
Volume 2
David Tong
Published online:

26 June 2025

Print publication:

19 June 2025
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There are four forces in our universe. Two act only at the very smallest scales and one only at the very biggest. For everything inbetween, there is electromagnetism. The theory of electromagnetism is described by four gloriously simple and beautiful vector calculus equations known as the Maxwell equations. These are the first genuinely fundamental equations that we meet in our physics education and they survive, essentially unchanged, in our best modern theories of physics. They also serve as a blueprint for what subsequent laws of physics look like. This textbook takes us on a tour of the Maxwell equations and their many solutions. It starts with the basics of electric and magnetic phenomena and explains how their unification results in waves that we call light. It then describes more advanced topics such as superconductors, monopoles, radiation, and electromagnetism in matter. The book concludes with a detailed review of the mathematics of vector calculus.

4 - Vector and Matrix Calculus
from Part I - Matrix Methods
Kevin W. Cassel, Illinois Institute of Technology
Book:

Matrix, Numerical, and Optimization Methods in Science and Engineering

Published online:

18 February 2021

Print publication:

04 March 2021, pp 218-252
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Summary

Vector and matrix calculus provides a powerful set of tools for analying and manipulating scalars, vectors, and tensors in continuum mechanics.This includes transformations between coordinate systems and provides the foundation for optimization methods.

Appendix B - Vector Calculus: Curvilinear Coordinates
Joel Franklin, Reed College, Oregon
Book:

Mathematical Methods for Oscillations and Waves

Published online:

02 April 2020

Print publication:

05 March 2020, pp 238-252
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Summary

A review of basic vector calculus expressions, like the gradient, divergence, curl and Laplacian, in cylindrical, spherical and more general coordinate systems.

6 - Waves in Three Dimensions
Joel Franklin, Reed College, Oregon
Book:

Mathematical Methods for Oscillations and Waves

Published online:

02 April 2020

Print publication:

05 March 2020, pp 146-180
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Summary

An exploration of the wave equation and its solutions in three dimensions.The chapter's mathematical focus is on vector calculus, enough to understand and appreciate the harmonic functions that make up the static solutions to the wave equation.

6 - Multivariable Calculus
Bruce F. Torrence, Randolph-Macon College, Virginia, Eve A. Torrence, Randolph-Macon College, Virginia
Book:

The Student's Introduction to <I>Mathematica</I> and the Wolfram Language

Published online:

01 April 2019

Print publication:

16 May 2019, pp 249-334
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Summary

Using Mathematica and the Wolfram Language to engage with calculus in a mutivariate setting. Includes curves, surfaces, plotting, differentiation, optimization, integrals, vector fields, line and surface integrals.

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