Plan for PLMPS

Pony Lectures on Mathematics for Physics Student (PLMPS) aim at providing an applicable mathematical textbook for undergraduates major in physics. The textbook should include all the main mathematical tools required in all the elementary and advanced physics courses, and should neglect some involving knowledge they’ll seldom make use of. Beside, PLMPS should be able to help the students to appreciate the inherent beauty of mathematics and physics.

PLMPS can be divided into four volumes. The first two volumes are designed for freshman and sophomore students, respectively. These are, actually, almost sufficient to meet the basic demand of all undergraduate courses. Volume 3 and 4 are for those who are ambitious enough to conquer some advanced subjects, or desire to attend a research group and publish some paper as an undergraduate student. They will introduce something more challenging and exciting.

The detail plan of PLMPS is as below.

Volume I: Basics Calculus & Linear Algebra
The first part will include most of the knowledge that will be taught in standard “Advanced Mathematics” courses. However, I intend to greatly emphasize something that will be of great use in physics courses, like “gradient, divergence & curl” and practical ODE skills. Meanwhile, a lot are to be omitted, such like most involving proving of rarely-used theorems. A brief introduction to complex numbers will be attached in the end of this part, in case the students will need them in electromagnetism. The second part will include a lot more than ordinary algebra textbooks, and things that will play a significant role in analytic & quantum mechanics will be explained extraordinarily carefully. Furthermore, Tensor Algebra will be introduced (not in detail) as an addition to this part. This volume contain sufficient knowledge to be applied in the standard “Elementary/Ordinary Physics” course(s).

Volume II: PDE & Complex Analysis
Quite opposite to the widely-accepted sequence of teaching, I put PDE in front of Complex. Firstly, an average sophomore student is quite capable of mastering the elementary solution of PDE, like separation of variables, with but a basic knowledge of complex numbers. Secondly, comparing with complex analysis, a knowledge of PDE is more badly needed. Courses like electrodynamics and quantum mechanics will require a mastery of elementary PDE solving skills. The first part of this volume will include most content of “Equations in Mathematical Physics” course, but parts that will require a better view of complex analysis, such like Bessel functions, will only be explained briefly. The second part will concentrate on the practical part of complex analysis, especially conformal transform, which will be used in some involving boundary problems. Knowledge that will be required to study the “special functions” will also be included, followed by a more thorough study of the special functions. Fourier & Laplace Transform and their application will be taught in great detail in this part. This volume will contain sufficient knowledge to be applied in the standard “Introductory Theoretical Physics (The Four Mechanics)” courses.

Volume III: Tensor Analysis & Non-linear Theory
The first part of the volume will explain tensor analysis in detail. This part is designed for applications in general relativity and cosmology, but it can easily be applies in other subjects like elasticity. The second part will give an introductory explanation to non-linear science and chaos.

Volume IV: Functional Analysis & Group Theory
The first part is an entirely utilitarian introduction of functional analysis. It will contain sufficient knowledge for its application in most situations an undergraduate can come across. The second part will systematically introduce the group theory, and will contain sufficient knowledge for its application in “Advanced Quantum Mechanics” and “Solid-state Physics”.

This is not the final version of my plan, for the purpose of releasing it is for your precious suggestion about how to adjust it.