---
date: 2022-04-06 18:52
modification date: Wednesday 6th April 2022 18:52:36
title: "differential forms"
aliases: [differential form]
---

Last modified date: <%+ tp.file.last_modified_date() %>

---
- Tags: 
	- #geomtop/differential-geometry 
- Refs:
	- #todo/add-references
- Links:
	- [Stokes theorem](Stokes%20theorem.md)

---

# Differential Forms

- An $n\dash$form on $\RR^n$: $\omega=f(\mathbf{x}) d x_{1} \wedge \cdots \wedge d x_{n}$
- A $k\dash$form on $\RR^n$ for $k<n$: $=\Sum_{i_{1}<\cdots<i_{k}} f_{I} \,\, d x_{i_{1}} \wedge \cdots \wedge d x_{i_{k}}$ where $f_I$ are smooth functions
- Differentiating a function: $$d f=\sum_{j=1}^{n} \frac{\partial f}{\partial x_{j}} d x_{j}$$
- Differentiating a form: $$d \omega=\sum_{I} d f_{I} \wedge d \mathbf{x}_{I}=\sum_{I} \sum_{j=1}^{n} \frac{\partial f_{I}}{\partial x_{j}} d x_{j} \wedge d x_{i_{1}} \wedge \cdots \wedge d x_{i_{k}}$$

# Example computations

- Differentiating a form:
 $$\begin{aligned}
d(y\dx + x\dy ) &= df_{0, 1} \wedge dx_{0, 1} + df_{1,0} \wedge dx_{1,0}\\
&= d(x) \wedge dy + d(y)\wedge dx \\
&= \qty{\dd{}{x}(y)\dx + \dd{}{y} (y)\dy } \wedge dx + \qty{\dd{}{x} (x) \dx + \dd{}{y} (x) \dy}\wedge dy \\
&= \qty{0 + 1\dy}\wedge \dx + (1\dx + 0)\wedge \dy \\
&= 1\dy\wedge dx + 1\dx\wedge \dy \\
&= (-1)\dx\wedge\dy + 1\dx\wedge\dy \\
&= 0
\end{aligned}$$
- Differentiating a form: 
$$\begin{aligned}
d(-y\dx + x\dy ) &= df_{0, 1} \wedge dx_{0, 1} + df_{1,0} \wedge dx_{1,0}\\
&= d(x) \wedge dy + d(-y)\wedge dx \\
&= \qty{\dd{}{x}(-y)\dx + \dd{}{y} (-y)\dy } \wedge dx + \qty{\dd{}{x} (x) \dx + \dd{}{y} (x) \dy}\wedge dy \\
&= \qty{0 + (-1) \dy}\wedge \dx + (1\dx + 0)\wedge \dy \\
&= (-1)\dy\wedge dx + (1)\dx\wedge \dy \\
&= 2 dx\wedge \dy
\end{aligned}$$
- Differentiating a function:
  $$\begin{aligned}
d(\arctan(y/x)) &= \dd{}{x} \arctan(y/x) \dx + \dd{}{y}\arctan(y/x) \dy \\
&= {1\over 1+ (y/x)^2} {-y\over x^2} \dx + {1\over 1+ (y/x)^2} {1\over x}\dy \\
&= {x^2 \over x^2 + y^2} {-y\over x^2} \dx + {x^2\over x^2 + y^2} {1\over x}\dx \\
&=  \qty{-{y\over x^2 + y^2} \dx + {x\over x^2 + y^2}\dy}
\end{aligned}$$
	- Checking that $d^2 = 0$:
 $$\begin{aligned}
d^2(\arctan(y/x)) 
&= d\qty{-{y\over x^2 + y^2} \dx + {x\over x^2 + y^2}\dy} \\
&\da d\qty{f_{1, 0} \dx + f_{0, 1} \dy} \\
&\da \qty{\dd{}{x}f_{1, 0} \dx + \dd{}{y}f_{1, 0} \dy} \wedge \dx + 
\qty{\dd{}{x}f_{0, 1} \dx + \dd{}{y}f_{0, 1} \dy} \wedge \dy \\
&= \qty{{2xy \over (x^2+y^2)^2}\dx + { y^2-x^2 \over (x^2+y^2)^2}\dy}\wedge dx + \qty{{y^2-x^2 \over (x^2+y^2)^2}\dx + {-2xy\over (x^2+y^2)^2}\dy}\wedge dy \\
&= {y^2-x^2 \over (x^2+y^2)^2}\dy\wedge \dx + {y^2-x^2 \over (x^2+y^2)^2}\dx \wedge \dy \\
&= -{y^2-x^2 \over (x^2+y^2)^2}\dx\wedge \dy + {y^2-x^2 \over (x^2+y^2)^2}\dx \wedge \dy \\
&= 0
\end{aligned}$$

- Differentiating higher $k\dash$forms:
  $$\begin{aligned}
d(x_1\dx_2 + x_3\dx_4 + x_5\dx_6) 
&= \qty{\sum \dd{}{x_i} (x_1) \dx_i}\wedge dx_2\\
&\qquad +\qty{\sum \dd{}{x_i} (x_3) \dx_i}\wedge dx_4 \\
&\qquad +
\qty{\sum \dd{}{x_i} (x_5) \dx_i}\wedge dx_6 \\
&= \dx_1\wedge \dx_2 + \dx_3 \wedge \dx_4 + \dx_5 \wedge \dx_6
\end{aligned}$$
- Wedging two 2-forms:

$$
\begin{aligned}
\omega&=a_{1} d x_{1}+a_{2} d x_{2} \text { and } \\
\eta&=b_{1} d x_{1}+b_{2} d x_{2} \in \Lambda^{1}\left(\mathbb{R}^{2}\right)^{*}=\left(\mathbb{R}^{2}\right)^{*} \implies \\ \\
\omega \wedge \eta &=\left(a_{1} d x_{1}+a_{2} d x_{2}\right) \wedge\left(b_{1} d x_{1}+b_{2} d x_{2}\right) \\
&=a_{1} b_{1} d x_{1} \wedge d x_{1}+a_{2} b_{1} d x_{2} \wedge d x_{1}+a_{1} b_{2} d x_{1} \wedge d x_{2}+a_{2} b_{2} d x_{2} \wedge d x_{2} \\
&=a_{1} b_{1} d \mathbf{x}_{(1,1)}+a_{2} b_{1} d \mathbf{x}_{(2,1)}+a_{1} b_{2} d \mathbf{x}_{(1,2)}+a_{2} b_{2} d \mathbf{x}_{(2,2)} \\
&=\left(a_{1} b_{2}-a_{2} b_{1}\right) d \mathbf{x}_{(1,2)} \\
&= \det\transp{\matt{\omega}{\eta}{\mid}{\mid}} = \det \matt{a_{1}}{a_2}{b_1}{b_2}
.
\end{aligned}
$$
- Integrating a form: let $\gamma(t) = \tv{1+t, -1+3t, 2t}$ for $t\in [0, 1]$ and $\omega = xy\dz$. Then
  $$
 \int_\gamma \omega = \int_0^1 \gamma^*\omega = \int_0^1 (1+t)(-1+3t)(2\dt) = 2
 $$