Differential Forms

Let $M\subset \mathbb{R}^n$ be an open set and $f:M\to \mathbb{R}^k$ a smooth map. Then for each $p\in M$ the Jacobian $f'(p)$ is a $k\times n$ matrix. The derivative $d_pf$ of $f$ at $p$ then is the linear map given by matrix multiplication with $f'(p)$:

\begin{align*}d_pf:\mathbb{R}^n &\to \mathbb{R}^k \\\\ d_pf(X)&=f'(p)X.\end{align*}

For example, if $f$ describes a regular surface in $\mathbb{R}^3$ (so $n=2$ and $k=3$), the derivative $df$ assigns to each point $p\in M$ a linear map $d_pf:\mathbb{R}^2 \to \mathbb{R}^3$ and $f'(p)$ is a $3\times 2$ matrix:

$$f'(u,v)=\left(\begin{array}{cc}|&|\\ f_u&f_v\\ |&|\end{array}\right).$$

The image of $d_pf$ is a two-dimensional subspace of $\mathbb{R}^3$ and is called the tangent plane of $f$ at $p$.

Suppose now $k=1$. Then $\omega:=df$ is a so-called 1-form on $M$, which means that $\omega$ assigns to each $p\in M$ a linear form $\omega_p:\mathbb{R}^n\to\mathbb{R}$. Let us define for $j\in \{1,\ldots,n\}$ the $j$th coordinate function $x_j:M\to \mathbb{R}$ by


Then for $p=(u_1\ldots,u_n)$ and

$$Y=\left(\begin{array}{c}y_1\\ \vdots \\y_n\end{array}\right).$$

we have

$$d_px_j(Y)= y_j.$$

Each 1-form $\omega$ on $M$ can be uniquely written as

$$\omega=a_1 \,dx_1 + \ldots + a_n \,dx_n.$$

with functions $a_1,\ldots a_n:M\to \mathbb{R}$ (they always will be assumed to be smooth). For example, if $f:M\to \mathbb{R}$ is a function then

$$df=\frac{\partial f}{\partial x_1} dx_1 +\ldots \frac{\partial f}{\partial x_n} dx_n.$$

A 2-form $\sigma$ on $M$ assigns to each $p\in M$ a skew-symmetric bilinear map

\begin{align*}\sigma_p:\mathbb{R}^n\times \mathbb{R}^n &\to \mathbb{R} \\\\ \sigma_p(X,Y)&=-\sigma_p(Y,X)\end{align*}

For two 1-forms $\omega$ and $\eta$ we define a 2-form $\omega \wedge \eta$ as

$$\omega \wedge \eta (X,Y)=\omega(X)\eta(Y)-\omega(Y)\eta(X).$$

For a 1-form $\omega$ given as above by functions $a_1,\ldots a_n$ we define the exterior derivative $d\omega$ as the 2-form

$$d\omega= da_1\wedge dx_1 +\ldots +da_n \wedge dx_n.$$

For example, for $n=2$ and $\omega=a\,du + b \,dv$ we have

$$d\omega= \left(\frac{\partial a}{\partial v}-\frac{\partial b}{\partial u}\right)du\wedge dv.$$

It is clear how to multiply a 1-form or a 2-form (from the left or from the right) by a function $f:M\to \mathbb{R}$. From these definitions we derive the following rules:

\begin{align*}\omega \wedge \eta&=-\eta \wedge \omega \\\\ d(f\omega)&=df\wedge \omega+f\,d\omega \\\\ d(\omega f)&=d\omega-\omega\wedge df.\end{align*}

If $\omega$ is a 1-form on $M$ and $\gamma:[a,b]\to \mathbb{R}^n$ is asmooth curve we define

$$\int_\gamma \omega := \int_a^b \omega(\gamma'(t))dt.$$

After defining $n$-forms we could give an $n$-dimensional version of the following definition, but for now let $M$ be a compact planar domain with smooth boundary $\partial M$ and let $\sigma=\rho\, du\wedge dv$ be a 2-form on $M$. Then we define

$$\int_M \sigma:= \int_M \rho\,\, du\,dv.$$

Now we can state a special case of the most important theorem that concerns differential forms.

Theorem of Stokes: Let $\omega$ be a 1-form on a compact planar domain$M$ with smooth boundary. Then

$$\int_M d\omega =\int_{\partial M} \omega.$$


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