# Analysis

:::info
Let $f:E\to\mathbb{R}$ be Riemann integrable on $E\subseteq\mathbb{R}^n$.
We use $\int_Ef\ d\mathbf{x}$ to represent the value of the integral.

Let $F=(f_1, \cdots, f_n):E\to\mathbb{R}^n$ such that each component is integrable, we write $\int_EF\ d\mathbf{x}$ as a vector $$\int_EF\ d\mathbf{x}=\left(\int_Ef_1\ d\mathbf{x}, \cdots, \int_Ef_n\ d\mathbf{x}\right).$$
:::

### ++**Coordinate System**++
Spherical Coordinate System: $\displaystyle\int_Ef\ d\mathbf{x}=\int_E f\ r^2\sin\theta\ dr\ d\theta\ d\varphi.$

Cylindrical Coordinate System: $\displaystyle\int_Ef\ d\mathbf{x}=\int_E f\ \rho\ dz\ d\rho\ d\varphi.$

### ++**Integration over $k$-forms**++

Let $M\subseteq\mathbb{R}^3$ be an oriented $k$-dimensional manifold, where $k\le 3$.
* A homeomorphism $\phi:U\subseteq\mathbb{R}^k\to M$ is called a **parametrization**.
* $C^\infty(M)=\{f:M\to\mathbb{R}\ |\ f\circ\phi\in C^\infty\}$.

Let $f\in C^\infty(M)$ and $\mathbf{f}=(f_1, f_2, f_3)$ where $f_i\in C^\infty(M)$ and set
* $\nabla=(\frac{\partial}{dx_1}, \frac{\partial}{dx_2}, \frac{\partial}{dx_3})$
* $d\mathbf{x}^1=(dx_1, dx_2, dx_3)$
* $d\mathbf{x}^2=(dx_2dx_3, dx_3dx_1, dx_1dx_2)$
* $d\mathbf{x}^3=(dx_1dx_2dx_3)$

| | 0-form | 1-form | 2-form | 3-form |
|:---------:|:-----------------------------:|:---------------------------------------------:|:--------------------------------------------:|:----------------------:|
| $\omega$ | $f$ | $\mathbf{f}\cdot d\mathbf{x}^1$ | $\mathbf{f}\cdot d\mathbf{x}^2$ | $f\cdot d\mathbf{x}^3$ |
| $d\omega$ | $\nabla f\cdot d\mathbf{x}^1$ | $(\nabla\times\mathbf{f})\cdot d\mathbf{x}^2$ | $(\nabla\cdot\mathbf{f})\cdot d\mathbf{x}^3$ | - |
| $J$ | - | $\phi'$ | $\phi_{x_1}\times\phi_{x_2}$ | $\det D\phi\ (=1)$ |

:::spoiler More info
> We use the dot product instead of the summation notation.
> Let $M\subseteq\mathbb{R}^n$ be an oriented $k$-dimensional manifold, where $k\le n$.
> Given a parametrization $\phi:U\subseteq\mathbb{R}^k\to M$ of $M$.
> * A **simple $k$-form** is just like $\omega=f\ d\mathbf{x}^I$, where $f\in C^\infty(M)$ and $d\mathbf{x}^I=dx^{n_1}dx^{n_2}\cdots dx^{n_k}$.
> * The **derivative** of a simple $k$-form $f\ d\mathbf{x}^I$ is defined by $$d\omega=\sum_{i=1}^n\frac{\partial f}{\partial x^i}dx^id\mathbf{x}^I=\nabla f\cdot (dx^1, \cdots, dx^n)d\mathbf{x}^I.$$
> * The **integral** of a simple $k$-form $f\ d\mathbf{x}^I$ is defined by $$\int_M\omega=\pm\int_U (f\circ\phi)\frac{\partial(x^{n_1}, \cdots, x^{n_k})}{\partial(u^1, \cdots, u^k)}\ du^1\cdots du^k$$ where $\frac{\partial(x^{n_1}, \cdots, x^{n_k})}{\partial(u^1, \cdots, u^k)}$ is the Jacobian and the sign depend on the orientation of $M$.
> * A **$k$-form** is a summation of simple $k$-forms.
:::
Note that the identity is always a parameterization of a 3-manifold.

* **The integral of a simple form** $\omega=\alpha\cdot d\mathbf{x}^k$ is defined by $$\int_M\omega=\int_U(\alpha\circ\phi)\cdot J\ d\mathbf{x},$$ where $\phi:U\subseteq\mathbb{R}^k\to M$ is a parametrization.
* The integral of a 0-form $\omega=f$ is defined by $\int_M\omega=f(\phi(0))$.
* **Stokes' Theorem**. $$\int_Md\omega=\int_{\partial M}\omega$$ where $\partial M$ is the boundary of $M$.

# Classical Physics

#### Notation:
* $\gamma\subseteq\mathbb{R}^3$ be an oriented $1$-dimensional manifold.
* $S\subseteq\mathbb{R}^3$ be an oriented $2$-dimensional manifold.
* $V\subseteq\mathbb{R}^3$ be an oriented $3$-dimensional manifold.
* $M\subseteq\mathbb{R}^n$ be an oriented $k$-dimensional manifold.
* $I=[a, b]\subseteq\mathbb{R}$ be a time interval.
* **Speed of Light** be a constant $c=299792458$.

#### Definition:
* **Scalar field** is a real-valued function $\mathbf{S}:M\subseteq\mathbb{R}^n\to\mathbb{R}$.
* **Vector field** is a vector-valued function $\mathbf{F}:M\subseteq\mathbb{R}^n\to\mathbb{R}^m$.
* **Work** of $\mathbf{F}$ on $\gamma$ is defined by $W_\mathbf{F}=\int_\gamma\mathbf{F}\cdot d\mathbf{x}^1$.
* **Flux** of $\mathbf{F}$ on $S$ is defined by $\Phi_\mathbf{F}=\int_S\mathbf{F}\cdot d\mathbf{x}^2$.
* **Potential** of $\mathbf{F}$ is a functon $\Psi$ such that $$\mathbf{F}\cdot d\mathbf{x}^{k+1}=d(\Psi\cdot d\mathbf{x}^k).$$

Particularly,
* $\mathbf{F}$ is said to be **conservative** if there exists $\Psi\in C^\infty(M)$ such that $$-\mathbf{F}\cdot d\mathbf{x}^1=\nabla\Psi\cdot d\mathbf{x}^1=d\Psi.$$
* **Potential difference** of a conservative field $\mathbf{F}$ along $\gamma$ is defined by $$\Delta\Psi_\gamma=-W_\mathbf{F}=\int_\gamma\nabla\Psi\cdot d\mathbf{x}^1=\int_{\partial \gamma}\Psi=\Psi(r(b))-\Psi(r(a))$$ where $r:I\to\gamma$ is a parametrization and $-\mathbf{F}=\nabla\Psi$.

### ++**Time-Invariant System**++
* **Physical Constant** depend on different systems

| Constant | Electric | Gravity |
|:--------:|:--------------------:|:----------------------:|
| $k$ | $c^2\times 10^{-7}$ | $-6.6743\times10^{-11}$ |

* **Density** is a scalar field $\rho:V\to\mathbb{R}$.
* **Quantity** of $\rho$ at time $t$ is defined by $Q=\int_V\rho\cdot d\mathbf{x}^3$.

:::info
The **gradient field** generated by density $\rho$ is defined by
$$\mathbf{G}:V\subseteq\mathbb{R}^3\to\mathbb{R}^3:\mathbf{a}\mapsto k\int_V\rho(\mathbf{x})\cdot\frac{\mathbf{x}-\mathbf{a}}{\|\mathbf{x}-\mathbf{a}\|^3}d\mathbf{x}.$$
:::
Every gradient field $\mathbf{G}$ is conservative and its potential can be given by $$\Psi:V\subseteq\mathbb{R}^3\to\mathbb{R}:\mathbf{a}\mapsto k\int_V\frac{\rho(\mathbf{x})}{\|\mathbf{x}-\mathbf{a}\|}d\mathbf{x}.$$
:::spoiler Proof
> Fixed $\mathbf{x}\in V$, then $\nabla\frac{1}{\|\mathbf{x}-\mathbf{a}\|}=-\frac{\mathbf{x}-\mathbf{a}}{\|\mathbf{x}-\mathbf{a}\|^3}$.
> It follows that $$\nabla\Psi(\mathbf{a})=k\int_V\nabla\left(\frac{\rho(\mathbf{x})}{\|\mathbf{x}-\mathbf{a}\|}\right)d\mathbf{x}=-\mathbf{G}(\mathbf{a}).$$ Note that $\nabla$ only acts on variable $\mathbf{a}$.
:::
:::spoiler Example 1
> **Cylindrical Coordinate System**
> Given $V=[0, \varepsilon]\times[0, 2\pi]\times[z_1, z_2]$ be a line with $\varepsilon\to 0$.
> Let $\mathbf{a}=(0, 0, c)\in\mathbb{R}^3$ and $\mathbf{x}=(r, \phi, z)\in V$, then $\mathbf{x}-\mathbf{a}\to(0, 0, z-c)$.
> Suppose that $\rho$ is a constant, then $Q=\int_V\rho\cdot d\mathbf{x}^3=(z_2-z_1)\varepsilon^2\pi$.
> In conclusion, $$\begin{matrix}\mathbf{G}(\mathbf{a}) &=& \displaystyle k\rho\int_V\frac{\mathbf{x}-\mathbf{a}}{\|\mathbf{x}-\mathbf{a}\|^3}d\mathbf{x} \\&\approx& \displaystyle k\pi\varepsilon^2\rho\int_{z_1}^{z_2}\frac{(0, 0, z-c)}{(z-c)^3}\ dz \\&=& k\pi\varepsilon^2\rho\left(0, 0, \frac{1}{z_1-c}-\frac{1}{z_2-c}\right) \\&=& \left(0, 0, \frac{kQ}{(z_1-c)(z_2-c)}\right).\end{matrix}$$
:::
:::spoiler Example 2
> **Cylindrical Coordinate System**
> Given $V=[0, \varepsilon]\times[0, 2\pi]\times\mathbb{R}$ be a line with $\varepsilon\to 0$.
> Let $\mathbf{a}=(a, 0, 0)\in\mathbb{R}^3$ and $\mathbf{x}=(r, \phi, z)\in V$, then $\mathbf{x}-\mathbf{a}\to(a, 0, z)$.
> Suppose that $\rho$ is a constant, then $$\begin{matrix}\mathbf{G}(\mathbf{a}) &=& \displaystyle k\rho\int_V\frac{\mathbf{x}-\mathbf{a}}{\|\mathbf{x}-\mathbf{a}\|^3}d\mathbf{x} \\&\approx& \displaystyle k\pi\varepsilon^2\rho\int^\infty_{-\infty}\frac{(a, 0, z)}{(a^2+z^2)^{3/2}}\ dz \\&=& \left(\frac{2k\pi\varepsilon^2\rho}{a}, 0, 0\right).\end{matrix}$$
> If we write $\lambda=\pi\varepsilon^2\rho$ as linear density distribution, then $$\mathbf{G}(\mathbf{a})=\left(\frac{2k\lambda}{a}, 0, 0\right).$$
:::

###

* **Velocity field** is a vector field $\mathbf{v}:V\to\mathbb{R}^3$.
* **Flux density** is a vector field $\mathbf{J}=\rho\mathbf{v}$.

:::info
The **curl field** generated by flux density $\mathbf{J}$ is defined by
$$\mathbf{C}:V\subseteq\mathbb{R}^3\to\mathbb{R}^3:\mathbf{a}\mapsto \frac{k}{c^2}\int_V\mathbf{J}(\mathbf{x})\times\frac{\mathbf{x}-\mathbf{a}}{\|\mathbf{x}-\mathbf{a}\|^3}d\mathbf{x}.$$
:::
Similarly, the potenital of $\mathbf{C}$ can be given by $$\Psi:V\subseteq\mathbb{R}^3\to\mathbb{R}:\mathbf{a}\mapsto \frac{k}{c^2}\int_V\frac{\mathbf{J}(\mathbf{x})}{\|\mathbf{x}-\mathbf{a}\|}d\mathbf{x}.$$ In this case, $\nabla\times\Psi=\mathbf{C}$.

### ++**Time-variant System**++

Let $\mathbf{F}:M\to\mathbb{R}^m$ be a vector field.
We extend $\mathbf{F}$ to be a **time-variant field** $\overline{\mathbf{F}}:M\times I\to\mathbb{R}^m$.
Whenever $t\in I$, we set
$$\mathbf{F}_t:M\to\mathbb{R}^m:\mathbf{a}\mapsto\overline{\mathbf{F}}(\mathbf{a}, t)$$ or just write it as $\mathbf{F}$ if there is no confusion.
The $\nabla$ operator here does not act on the time dimension.

**Field Equations**.
Let $\partial V=S$ be a closed surface and $\partial S=\gamma$ be a closed curve.
| | 2-3-form | 1-2-form |
| -------- | -------- | -------- |
| gradient | $\begin{matrix}\nabla\cdot\mathbf{G}&=&4\pi k\rho\\\int_S\mathbf{G}\cdot d\mathbf{x}^2&=&4\pi kQ\end{matrix}$ | $\begin{matrix}\nabla\times\mathbf{G}&=&-\frac{\partial\mathbf{C}}{\partial t}\\\int_\gamma\mathbf{G}\cdot d\mathbf{x}^1&=&-\frac{d\Phi_\mathbf{C}}{dt}\end{matrix}$ |
| curl | $\begin{matrix}\nabla\cdot\mathbf{C}&=&0\\\int_S\mathbf{C}\cdot d\mathbf{x}^2&=&0\end{matrix}$ | $\begin{matrix}\nabla\times\mathbf{C}&=&\frac{1}{c^2}(\frac{\partial\mathbf{G}}{\partial t}+4\pi k\mathbf{J})\\\int_\gamma\mathbf{C}\cdot d\mathbf{x}^1&=&\frac{1}{c^2}(\frac{d\Phi_\mathbf{G}}{dt}+4\pi k\Phi_\mathbf{J})\end{matrix}$ |

If $\frac{\partial\mathbf{C}}{\partial t}\ne 0$ or $\frac{\partial\mathbf{G}}{\partial t}\ne 0$, then the potential of $\mathbf{G}$ and $\mathbf{C}$ are **not exists**.

**Continuity Equation.**
$$\frac{\partial\rho}{\partial t}+\nabla\cdot\mathbf{J}=0.$$

Fixed $t\in I$, then by the continuity equation, $$\Phi_\mathbf{J}=\int_{\partial V}\mathbf{J}\cdot d\mathbf{x}^2=\int_V(\nabla\cdot\mathbf{J})\cdot d\mathbf{x}^3=-\int_V\frac{\partial\rho}{\partial t}\cdot d\mathbf{x}^3=-\frac{d}{dt}\int_V\rho\cdot d\mathbf{x}^3,$$ which is the **conservation law**.

**Linearly Polarized Waves.**
Set
* $\rho=\mathbf{J}=0$ and $\omega\in\mathbb{R}$.
* $f:\mathbb{R}\times I\to\mathbb{R}$ of class $C^2$

Then $$\mathbf{G}=\left(0, f\left(\frac{\omega}{c}x_1-\omega t\right), 0\right),\quad \mathbf{C}=\left(0, 0, \frac{1}{c}f\left(\frac{\omega}{c}x_1-\omega t\right)\right)$$ is a solution for field equations.
:::spoiler More info
> Set $\mathbf{G}=(G_1, G_2, G_3)$ and $\mathbf{C}=(C_1, C_2, C_3)$ of class $C^2$.
> Suupose that $\mathbf{G}$ and $\mathbf{C}$ are invariant along $x_2$-axis and $x_3$-axis.
> Then
$$\nabla\times\mathbf{G}=\left(0,0,\frac{\partial G_2}{\partial x_1}\right)=-\frac{\partial\mathbf{C}}{\partial t},\quad\nabla\times\mathbf{C}=\left(0,-\frac{\partial C_3}{\partial x_1}, 0\right)=\frac{1}{c^2}\frac{\partial\mathbf{G}}{\partial t}.$$ It follows that $0=\frac{\partial G_1}{\partial t}=\frac{\partial G_3}{\partial t}=\frac{\partial C_1}{\partial t}=\frac{\partial C_2}{\partial t}$.
> Since $G_2$ and $C_3$ have continuous second partial derivatives, we have $$\frac{\partial^2 G_2}{\partial x_1^2}=\frac{\partial}{\partial t}\frac{\partial C_3}{\partial x_1}=\frac{1}{c^2}\frac{\partial^2 G_2}{\partial t^2},\quad\frac{\partial^2 C_3}{\partial x_1^2}=\frac{1}{c^2}\frac{\partial^2 C_3}{\partial t^2}.$$
:::

### ++**Related Concepts**++

Suppose that $(V, \gamma, \pi:V\to \gamma, S)$ is a **fiber bundle**.
* $\pi^{-1}(p)$ is homeomorphic to $S$ and is called the **fiber** of $p\in\gamma$.
:::spoiler More info
Imagine that $V$ is a conductor, $\gamma$ is the central axis of $V$, and $S$ is the section of $V$:
* The **bundle projection** $\pi:V\to\gamma$ is a continuous surjection.
* The **base space** $\gamma$ is connected.
* For all $x\in\gamma$,
* there is an open neighborhood $U_x$ of $x$ in $\gamma$
* there is a homeomorphism $\phi_x:\pi^{-1}(U_x)\to U_x\times S$
* $p_x:U_x\times S\to U_x:(y, z)\to y$ is the projection.
* $\pi=p_x\circ\phi_x$ on $U_x\times S$.
:::
###

If $\mathbf{J}$ is a constant on $V$ and $S_0$ be a surface, then $$L=\frac{\Phi_\mathbf{C}\text{ on }S_0}{\Phi_\mathbf{J}}=\frac{\int_{S_0}\mathbf{C}\cdot d\mathbf{x}^2}{\int_S\mathbf{J}\cdot d\mathbf{x}^2}$$ is called the **inductance** on $S_0$.
* If $\gamma_0=\partial S_0$, then $W_\mathbf{G}=-\frac{d\Phi_{\mathbf{C}}\text{ on }S_0}{dt}=-L\frac{d\Phi_{\mathbf{J}}}{dt}$ by the field equation.
* If $V\approx\partial S_1$ for some surface $S_1$, then $L$ is called the mutual-inductance.
* If $V\approx\partial S_0$, then $L$ is called the self-inductance.

Suppose that $\mathbf{G}$ and $\mathbf{C}$ on $V$ are time-invariant.
Set $\Delta\Psi_\gamma$ to be the potential difference of $\mathbf{G}$.
* **Capacitance** is defined by $C=\int_V\rho\cdot d\mathbf{x}^3/\Delta\Psi_\gamma=Q/\Delta\Psi_\gamma$.
* If $\Phi_\mathbf{J}$ is a constant on all fiber $\pi^{-1}(p)$, then $R=\Delta\Psi_\gamma/\Phi_\mathbf{J}$ is a constant called **resistance** and $V$ is said to be **Ohmic**.

Let $p\in V$ be a point with a poined quantity $q\in\mathbb{R}$.
* **Poynting vector** is defined by $\frac{c^2}{4\pi k}\mathbf{G}\times\mathbf{C}$.
* **Potential energy** of $p$ is defined by $\Delta U=q\Delta\Psi_\gamma$.
:::info
Let $\mathbf{G}$ be a gradient field, $\mathbf{C}$ be a curl field, $\mathbf{v}$ be a velocity field.
The **Force** on $p$ is given by $$q(\mathbf{G}+\mathbf{v}\times\mathbf{C})$$
:::

# Mechanics

### Linear System
* **Mass Density** $\rho$ in *Linear Systems* is generated by **gracity**.
* **Mass** $m$ is the quantity in linear system.
* **Gravitational field** $g$ is the field generated by a mass density.

### Rotation System
* The density in *Rotation Systems* is defined by $\rho r^2$, where $\rho$ is the mass density and $r$ is the distance from the axis of rotation.
* **Inertia** $I$ is the quantity in rotation system.

Let $q$ be the quantity in a system.
* **Momentum** is defined by $p=\|q\mathbf{v}\|$.
* **Kinetic Energy** is is defined by $K=\frac{1}{2}q\|\mathbf{v}\|^2$.

# Electricity

Suppose that $(V, \gamma, \pi:V\to\gamma, S)$ is a Ohmic fiber bundle.

| | Series | Parallel |
|:----------:|:----------------:|:----------------:|
| Definition | $\gamma=\bigcup_i\gamma_i$ | $S=\bigcup_iS_i$ |
| $W_\mathbf{G}$ | $\sum_iW_\mathbf{G}\text{ on }\gamma_i$ | $W_\mathbf{G}$ |
| $\Phi_\mathbf{J}$ | $\Phi_\mathbf{J}$ | $\sum_i\Phi_\mathbf{J}\text{ on }S_i$ |
| Capacitance | $\frac{1}{C}=\sum_i\frac{1}{C_i}$ | $C=\sum_iC_i$ |
| Resistance | $R=\sum_i R_i$ | $\frac{1}{R}=\sum_i\frac{1}{R_i}$ |
| Inductance on $S_0$ | $L=\sum_i L_i$ | $\frac{1}{L}=\sum_i\frac{1}{L_i}$ |

### RC Circuits
1. $\gamma=\gamma_0\cup \gamma_1\cup \gamma_2$ is a closed curve.
2. $\gamma_0$ has a constant work $W$ when charging.
3. $\gamma_1$ has a constant caparacitance $C$.
4. $\gamma_2$ has a constant resistance $R$.
5. Ignore $\frac{d\Phi_\mathbf{G}}{dt}$ and $\frac{d\Phi_\mathbf{C}}{dt}$.

Let $q$ be the quantity on $\gamma_1$.
| | Charging | Discharging |
|:--------------------------:| -------- | ----------- |
| When $t=0$ | $q=0$ | $q=Q$ |
| $q$ | $WC(1-e^{-t/RC})$ | $Qe^{-t/RC}$ |
| $\Phi_\mathbf{J}=\frac{dq}{dt}$ | $\frac{W}{R}e^{-t/RC}$ | $-\frac{Q}{RC}e^{-t/RC}$ |
:::spoiler Proof
> **Charging**
> By continuity equation, $\Phi_\mathbf{J}=\frac{dq}{dt}$ and hence $\Delta\Psi_{\gamma_2}=R\Phi_\mathbf{J}=R\frac{dq}{dt}$.
> Since $\gamma$ is closed, $0=W_\mathbf{G}=W-\frac{q}{C}-R\frac{dq}{dt}$.
> Then $\frac{dq}{dt}=\frac{W}{R}-\frac{q}{RC}$ and $q=c_1e^{-t/RC}+WC$ for some $c_1$.
> Since $q=0$ when $t=0$, we conclude that $$q=WC(1-e^{-t/RC}).$$
> **Disharging**
> Note that $0=W_\mathbf{G}=-\frac{q}{C}-R\frac{dq}{dt}$.
> Then $q=c_1e^{-t/RC}$ for some $c_1$.
> Since $q=Q$ when $t=0$, $$q=Qe^{-t/RC}.$$
:::

### RLC Series Circuits
1. $\gamma=\gamma_0\cup\gamma_1\cup \gamma_2$ is a closed curve.
2. $\gamma_0$ has a constant caparacitance $C$.
3. $\gamma_1$ is closed and has a constant inductance $L$.
4. $\gamma_2$ has a constant resistance $R$.
5. Ignore $\frac{d\Phi_\mathbf{G}}{dt}$ and $\frac{d\Phi_\mathbf{C}}{dt}$.

Let $q$ be the quantity $q$ on $\gamma_0$.
Suppose that $q=Q$ and $\Phi_\mathbf{J}=0$ when $t=0$.
If $R^2-4L/C<0$, then $$q=Qe^{\alpha t}\cos\beta t$$ where $\alpha=\frac{-R}{2L}$ and $\beta=\frac{\sqrt{4L/C-R^2}}{2L}$.
:::spoiler Proof
> Since $\gamma_1$ is closed, $W_{\mathbf{G}_1}=-L\frac{d\Phi_\mathbf{J}}{dt}=-L\frac{d^2q}{dt^2}$.
> Since $\gamma$ is closed, $0=W_\mathbf{G}=-\frac{q}{C}-L\frac{d^2q}{dt^2}-R\frac{dq}{dt}$.
> Since $\alpha\pm i\beta$ are roots of $Lx^2+Rx+\frac{1}{C}$, $$q=c_1e^{\alpha t}\cos\beta t+c_2e^{\alpha t}\sin\beta t$$ for some $c_1$ and $c_2$.
> Since $q=Q$ and $\Phi_\mathbf{J}=\frac{dq}{dt}=0$ when $t=0$, we have $c_1=Q$ and $c_2=0$.
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