AmericanTopography

Studies Directory

Table of Contents

  1. Table of Contents
  2. Introduction
  3. Math (Mxx)
    1. Fundamentals (MFx)
    2. Analysis (MMx)
    3. Algebra (MGx)
    4. Discrete (MDx)
    5. Dynamical Systems (MOx)
    6. Geometry (MRx)
    7. Number Theory (MNx)
    8. Topology (MTx)
    9. Numerical (MAx)
    10. Statistics & Probability (MSx)
  4. Physics (Pxx)
    1. Fundamentals (PFx)
    2. Electricity & Magnetism (PEx)
    3. Quantum (PQx)
    4. Gravitational (PRx)
    5. Astrophysics (PAx)
    6. High Energy (PHx)
  5. Electrical & Computer Engineering (Exx)
    1. Fundamentals (EFx)
    2. Electronics (EEx)
    3. Digital Systems (EDx)
    4. Devices (EVx)
    5. Signal Processing (ESx)
    6. Controls (EKx)
    7. Communications (ECx)
    8. Power (EPx)
    9. Photonics & Lasers (ELx)
  6. Mechanical Engineering (Axx)
    1. TBD

Introduction

This is a long and uninformed list of study materials for different areas of the sciences. This list is by no means comprehensive. Each categroy has the following information:

  • Pre-requisites
  • A short description (usually the first line from wikipedia)
  • A list of resources

Each category additionally has a corresponding 3-character alpha-numeric for shorthands, which is useful for listing pre-requisites. Resources may or may not have comments detailing the difficulty or quiks of the resource.

Enjoy.


Math (Mxx)

Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes.

Resources:

  • The Elements, Euclid

Fundamentals (MFx)

1. Proofs & Logic (MF1)

Logic is the study of correct reasoning.

PR: None

Resources:

  • A Transition to Advanced Mathematics, Smith, Eggen, Andre
  • How to Prove It, Velleman

2. Real Analysis (MF2)

Real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions.

PR: MF1

Resources:

  • Principles of Mathematical Analysis, Rudin

    Standard, but difficult

  • Understanding Analysis, Abbott
  • Analysis I && II, Tao

3. Abstract Algebra (MF3)

Abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field.

PR: MF1

Resources:

  • Topics in Algebra, Herstein

    Standard approach

  • Algebra, Artin

    Geometric approach

  • Algebra, Lang

    Coprehensive and advanced

  • Abstract Algebra, Dummit, Foote

    Encyclopedic

  • Algebra Chapter 0, Aluffi
  • Contemporary Abstract Algebra, Gallian

4. Linear Algebra (MF4)

Linear algebra is the branch of mathematics concerning linear equations.

PR: MF1

Resources:

  • Linear Algebra and Its Applications, Strang
  • Linear Algebra Done Right, Axler

    Doesn’t cover determinants

  • Linear Algebra, Shilov
  • Linear Algebra and Geometry, Suetin, Kostrikin, Manin

    Advanced

  • Introduction to Linear ALgebra, Strang
  • Linear ALgebra, Freidberg, Insel, Spence

    Comprehensive

5. Complex Analysis (MF5)

Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers.

PR: MF1, MF2

Resources:

  • Complex Analysis: A First Course with Applications, Zill, Shanahan
  • Visual Complex Analysis, Needham
  • Complex Variables and Applications, Brown, Churchill
  • Fundamentals of Complex Analysis, Snider, Saff

Analysis (MMx)

1. Measure Theory (MM1)

In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude, mass, and probability of events.

PR: MF2, MF3

Resources:

  • Measure Theory, Halmos
  • An Introduction to Measure Theory, Tao
  • Real Analysis, Royden, Fitzpatrick
  • Real Analysis, Stein
  • Real and Complex Analysis, Rudin

2. Functional Analysis (MM2)

Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, inner product, norm, or topology) and the linear functions defined on these spaces and suitably respecting these structures.

PR: MM2, MT1

Resources:

  • Introduction to Functional Analysis with Applications, Kreyszig

    Easy introduction

  • Functional Analysis, Bachman, Narici

    Advanced

  • Functional Analysis, Kesavan
  • Functional Analysis, Rudin

Algebra (MGx)


Discrete (MDx)


Dynamical Systems (MOx)


Geometry (MRx)


Number Theory (MNx)


Topology (MTx)


Numerical (MAx)


Statistics & Probability (MSx)


Physics (Pxx)


Fundamentals (PFx)


Electricity & Magnetism (PEx)


Quantum (PQx)


Gravitational (PRx)


Astrophysics (PAx)


High Energy (PHx)


Electrical & Computer Engineering (Exx)


Fundamentals (EFx)


Electronics (EEx)


Digital Systems (EDx)


Devices (EVx)


Signal Processing (ESx)


Controls (EKx)


Communications (ECx)


Power (EPx)


Photonics & Lasers (ELx)


Mechanical Engineering (Axx)