TOC

• Chap1

  3 PI quantization of a Bose particle

  Suppose at $t_{i}$, the particle is at $x_{i}$. The probability amplitude of finding particle at $x_{f}$ at $t_{f}$ is
  $$\begin{equation}
  \left<x_{f},t_{f}|x_{i}...

• Chap2

  \fcolorbox{red}{}{Definition Homeomorphism}: Let $X_{1},X_{2}$ be topological spaces, a map $f:X_{1}\to X_{2}$ is a homeomorphism if it is continuous and has inverse $f^{-1}:X_{2}\to X_{1}$ which i...

• Chap3 Homology groups

  2.1 Simplexes

  \fcolorbox{red}{}{Definition (r-simplexes)}:
  r-simplex $\sigma _r=\left<p_{0}p_{1}\dots p_{r}\right>$ is defined as
  $$
  \begin{equation}
  \sigma _{r}=\left{x\in \mathbb{R}^{m}\mid ...

• Chap4 Homotopy groups

  1. Fundamental groups

  We first show that the loops with some fixed basepoint form a group under certain defined multiplication.
  \fcolorbox{red}{}{Definition Path} Let $X$ be a topological space ...

• Chap5 Manifold

  \fcolorbox{green}{}{Example RPn}
  $\mathbb{R}P^{n}$ is thee set of lines through the origin in $\mathbb{R}^{n+1}$. If $x\neq 0$, $x$ defines a line through origin. If $y\in \mathbb{R}^{n+1}$ defines...

• Chap6 De Rham Cohomology Groups

  6.1 Stokes' theorem

  Let the standard $n$-simplex $\bar{\sigma}_{r}$ given by
  $$
  \begin{equation}
  \bar{\sigma}{r} = \left{(x^{1},\dots,x^{r})\in \mathbb{R}^{r}\mid x^{\mu}\geq 0, \sum{\mu=1}^{r}x...

• Chap9

  9.1 Tangent Bundles

  \fcolorbox{red}{}{Definition}
  A tangent bundle $TM$ over an $m$-dimensional manifold $M$ is a collection of all tangent spaces of $M$:
  $$
  \begin{equation}
  TM\equiv \bigc...