Advanced Theoretical Physics

✍ Scribed by Nick Lucid

• Publisher: self-publ.
• Year: 2015
• Tongue: English
• Leaves: 524
• Category: Library

✦ Synopsis

[The book is self-published by the author.

Check out his Youtube and website, where you can purchase this book for any price you choose:

https://scienceasylum.com/

Of course you can still download it here.]

Are you an undergraduate physics student planning on attending graduate school? Are you a graduate physics student already, but you feel like you're missing something? Either way, this book could be a huge help! There are a lot of little bits of information that don’t get taught in undergraduate physics for one reason or another, but your graduate teachers still expect you to know it. That’s what this book is: all the super important bits you should know before entering graduate physics. This book is not intended for anyone without at least background in basic calculus and introductory physics, but if you like a challenge, then go for it!

Inside, you’ll find mathematical topics like vector calculus and tensor analysis as well as physics topics like electrodynamics, relativity, and quantum mechanics. Historical context is also given throughout the text to give you a deeper insight into each topic and to show how long it took the scientific community to develop the ideas, which can make you feel a little better about maybe not understanding it right away. You can find a table of contents and some.

You can also:

buy the book

watch on youtube

listen to the podcast

✦ Table of Contents

Preface
ix
1
Coordinate Systems
1
1.1
Cartesian
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.2
Polar and Cylindrical . . . . . . . . . . . . . . . . . . . . . . .
4
1.3
Spherical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
1.4
Bipolar and Elliptic . . . . . . . . . . . . . . . . . . . . . . . .
8
2
Vector Algebra
11
2.1
Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
2.2
Vector Operators . . . . . . . . . . . . . . . . . . . . . . . . .
12
3
Vector Calculus
19
3.1
Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
3.2
Del Operator
. . . . . . . . . . . . . . . . . . . . . . . . . . .
20
3.3
Non-Cartesian Del Operators
. . . . . . . . . . . . . . . . . .
24
3.4
Arbitrary Del Operator . . . . . . . . . . . . . . . . . . . . . .
33
3.5
Vector Calculus Theorems . . . . . . . . . . . . . . . . . . . .
36
The Divergence Theorem . . . . . . . . . . . . . . . . . . .
37
The Curl Theorem . . . . . . . . . . . . . . . . . . . . . .
39
4
Lagrangian Mechanics
45
4.1
A Little History... . . . . . . . . . . . . . . . . . . . . . . . . .
45
4.2
Derivation of Lagrange’s Equation . . . . . . . . . . . . . . . .
46
4.3
Generalizing for Multiple Bodies . . . . . . . . . . . . . . . . .
51
4.4
Applications of Lagrange’s Equation
. . . . . . . . . . . . . .
52
4.5
Lagrange Multipliers . . . . . . . . . . . . . . . . . . . . . . .
66
4.6
Applications of Lagrange Multipliers
. . . . . . . . . . . . . .
68
4.7
Non-Conservative Forces . . . . . . . . . . . . . . . . . . . . .
75
5
Electrodynamics
77
5.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
77
5.2
Experimental Laws . . . . . . . . . . . . . . . . . . . . . . . .
77
Coulomb’s Law . . . . . . . . . . . . . . . . . . . . . . . .
78
Biot-Savart Law. . . . . . . . . . . . . . . . . . . . . . . .
87
5.3
Theoretical Laws . . . . . . . . . . . . . . . . . . . . . . . . .
97
Amp´ere’s Law . . . . . . . . . . . . . . . . . . . . . . . . .
97
Faraday’s Law. . . . . . . . . . . . . . . . . . . . . . . . . 105
Gauss’s Law(s) . . . . . . . . . . . . . . . . . . . . . . . . 108
Amp´ere’s Law Revisited . . . . . . . . . . . . . . . . . . . 111
5.4
Unification of Electricity and Magnetism . . . . . . . . . . . . 114
5.5
Electromagnetic Waves . . . . . . . . . . . . . . . . . . . . . . 118
5.6
Potential Functions . . . . . . . . . . . . . . . . . . . . . . . . 123
Maxwell’s Equations with Potentials. . . . . . . . . . . . . 127
5.7
Blurring Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
6
Tensor Analysis
131
6.1
What is a Tensor?
. . . . . . . . . . . . . . . . . . . . . . . . 131
6.2
Index Notation
. . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.3
Matrix Notation . . . . . . . . . . . . . . . . . . . . . . . . . . 136
6.4
Describing a Space . . . . . . . . . . . . . . . . . . . . . . . . 141
Line Element . . . . . . . . . . . . . . . . . . . . . . . . . 141
Metric Tensor . . . . . . . . . . . . . . . . . . . . . . . . . 141
Raising and Lowering Indices . . . . . . . . . . . . . . . . 142
Coordinate Basis vs. Orthonormal Basis. . . . . . . . . . . 143
6.5 Really... What’s a Tensor?! . . . . . . . . . . . . . . . . . . . . 144
6.6
Coordinate Transformations . . . . . . . . . . . . . . . . . . . 149
6.7
Tensor Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . 154
7 Special Relativity 167
7.1 Origins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
7.2 Spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
Line Element . . . . . . . . . . . . . . . . . . . . . . . . . 170
Metric Tensor . . . . . . . . . . . . . . . . . . . . . . . . . 172
Coordinate Rotations . . . . . . . . . . . . . . . . . . . . . 173

✦ Subjects

self-published