15th March 2015

# On Differential Forms

Abstract. This article will give a very simple definition of $$k$$-forms or differential forms. It just requires basic knowledge about matrices and determinants. Furthermore a very simple proof will be given for the proposition that the double outer differentiation of $$k$$-forms vanishes.

MSC 2010: 58A10

# 1. Basic definitions.

We denote the submatrix of $$A=(a_{ij})\in R^{m\times n}$$ consisting of the rows $$i_1,\ldots,i_k$$ and the columns $$j_1,\ldots,j_k$$ with

$$[A]{\textstyle\!{\scriptstyle j_1\atop\scriptstyle i_1} \!{\scriptstyle\ldots\atop\scriptstyle\ldots} \!{\scriptstyle j_k\atop\scriptstyle i_k}} := \begin{pmatrix} a_{i_1j_1} & \ldots & a_{i_1j_k}\\ \vdots & \ddots & \vdots\\ a_{i_kj_1} & \ldots & a_{i_kj_k}\\ \end{pmatrix}$$

and its determinant with

$$A{\textstyle\!{\scriptstyle j_1\atop\scriptstyle i_1} \!{\scriptstyle\ldots\atop\scriptstyle\ldots} \!{\scriptstyle j_k\atop\scriptstyle i_k}} := \det [A]{\textstyle\!{\scriptstyle j_1\atop\scriptstyle i_1} \!{\scriptstyle\ldots\atop\scriptstyle\ldots} \!{\scriptstyle j_k\atop\scriptstyle i_k}}.$$

For example

$$A = \begin{pmatrix}a_{11}&a_{12}&a_{13}\\ a_{21}&a_{22}&a_{23}\\ \end{pmatrix}, \qquad A_{1,2}^{1,3} = a_{11}a_{23} - a_{21}a_{13}.$$

Suppose

$$H \in R^{n\times(n+1)}$$

and let

$$f,g\colon U\subseteq R^n\to R, \qquad U \text{ open},$$

be two functions which are two-times continuously differentiable. Then we call for a fixed $$k$$ the expression

$$f\,H_\alpha^{1\ldots k}, \qquad \alpha=\left(i_1,\ldots,i_k\right) \in\left\{1,\ldots,n\right\}^k,$$

a basic $$k$$-form or basic differential form of order $$k$$. It's a real function of $$n+k^2$$ variables. For $$k>n$$ the expression is defined to be zero. If $$f$$ also depends on $$\alpha$$ then

$$\sum_{1\le i_1\lt \cdots\lt i_k\le n} f_{i_1\ldots i_k} H{\textstyle\!{\scriptstyle1\atop\scriptstyle i_1} \!{\scriptstyle\ldots\atop\scriptstyle\ldots} \!{\scriptstyle k\atop\scriptstyle i_k}}$$

is called a $$k$$-form. It's a real function of $$n+kn$$ variables which is $$k$$-linear in the $$k$$ column-vectors of $$H$$.

For example for $$f\colon R\to R$$ and $$H\in R^{1\times1}$$ we have $$f(x),H$$. This is a linear function in $$H$$ and a possibly non-linear function in $$x$$.

# 2. Differentiation of $$k$$-forms.

For the differential form

$$\omega = f H^{1\ldots k}_\alpha, \qquad \alpha=\left(i_1,\ldots,i_k\right),$$

we define

$$d\omega := \sum_{\nu=1}^n {\partial f\over\partial x_\nu} H^{1\ldots k+1}_{\nu,\alpha}$$

as the outer differentiation of $$\omega$$. This is a $$(k+1)$$-form. It's a function of $$n+(k+1)n$$ variables.

The $$0$$-form

$$\omega = f, \qquad \left|\alpha\right|=k=0$$

yields

$$dw = \sum_{\nu=1}^n {\partial f\over\partial x_\nu} H^1_\nu \qquad(1)$$

which corresponds to $$\nabla f = \mathop{\rm grad}f$$.

In the special case $$k=\left|\alpha\right|=1$$ we get for

$$\omega = \sum_{i=1}^n f_i H^1_i$$

the result

$$d\omega = \sum_{i=1}^n \sum_{j=1}^n {\partial f_i\over\partial x_j} H^{1,2}_{j,i} = \sum_{i\lt j} \left({\partial f_i\over\partial x_j} - {\partial f_j\over\partial x_i}\right) H^{1,2}_{j,i}. \qquad(2)$$

This corresponds to $$\mathop{\rm rot} f$$.

Let hat ($$\hat{}$$) mean exclusion from the index list. The case $$k=n-1$$ for

$$\omega = \sum_{i=1}^n (-1)^{i-1} f_i\,H^{\: 1\ldots n-1\: }_{1\ldots\hat\imath\ldots n}$$

delivers

$$dw = \sum_{i=1}^n \sum_{\nu=1}^n (-1)^{i-1} {\partial f_i\over\partial x_\nu} H^{\: \: 1\ldots n}_{\nu,1\ldots\hat\imath\ldots n} = \sum_{i=1}^n {\partial f_i\over\partial x_\nu} H^{1\ldots n}_{1\ldots n} = \left(\sum_{i=1}^n{\partial f_i\over\partial x_i}\right) \det H.$$

This corresponds to $$\mathop{\rm div}f$$.

Theorem. For $$\omega = f H_\alpha^{1\ldots k}$$ we have

$$dd\omega = 0.$$

Proof: With

$$d\omega = \sum_{\nu=1}^n {\partial f\over\partial x_\nu} H_{\nu,\alpha}^{1\ldots k+1}$$

we get

$$dd\omega = \sum_{\nu=1}^n \sum_{\mu=1}^n {\partial^2 f\over\partial x_\nu\partial x_\mu} H_{\mu,\nu,\alpha}^{1\ldots k+2}$$

and this is zero, because

$$H_{\mu,\mu,\alpha}^{1\ldots k+2} = 0, \qquad H_{\mu,\nu,\alpha}^{1\ldots k+2} = -H_{\nu,\mu,\alpha}^{1\ldots k+2},$$

and

$${\partial^2f\over\partial x_\nu\partial x_\mu} = {\partial^2f\over\partial x_\mu\partial x_\nu}.$$

Application of this theorem to an $$0$$-form with an $$f\colon U\subseteq R^n\to R$$ and a $$1$$-form with an $$a\colon U\to R^n$$ reading (1) and then (2) yields

$$\mathop{\rm rot}\mathop{\rm grad} f = 0, \qquad \mathop{\rm div}\mathop{\rm rot} a = 0.$$

The second equation is only true for $$n=3$$ because

$${n\choose 2} = n \quad (n\in N) \qquad\Leftrightarrow\qquad n = 3.$$

Definition. Suppose

$$\phi\colon D\to E\subset R^n, \qquad D\subset\!\subset R^k,$$

is differentiable, its derivative denoted by $$\phi'$$, and

$$f\colon E\to R.$$

For the differential form $$\omega = f H^{1\ldots k}_\alpha$$ we define the back-transportation as

$$\phi^*\omega := (f\circ\phi) \, (\phi')^{1\ldots k}_{\alpha}$$

and the integral over $$k$$-forms as

$$\int_\phi \omega := \int_D \phi^*\omega.$$

For example the case $$k=1$$,

$$\omega = \sum_{i=1}^n f_i H^1_i$$

gives

$$\phi^*\omega = \sum_{i=1}^n (f_i\circ\phi) \, (\phi')_i^1 .$$

# 3. The outer product of differential forms.

Suppose

$$H\in R^{n\times(n+1)}, \qquad k+m\leq n.$$

For the two differential forms

$$\omega = \sum_{1\le i_1\lt \cdots\lt i_k\le n} f_{i_1\ldots i_k} H{\textstyle\!{\scriptstyle1\atop\scriptstyle i_1} \!{\scriptstyle\ldots\atop\scriptstyle\ldots} \!{\scriptstyle k\atop\scriptstyle i_k}}$$

and

$$\lambda = \sum_{1\le j_1\lt \cdots\lt j_m\le n} g_{j_1\ldots j_m} H{\textstyle\!{\scriptstyle k+1\atop\scriptstyle j_1} \!{\scriptstyle\ldots\atop\scriptstyle\ldots} \!{\scriptstyle k+m\atop\scriptstyle j_m}}$$

the outer product is defined as

$$w\land\lambda := \sum _{\scriptstyle1\le i_1\lt \cdots\lt i_k\le n\atop \scriptstyle1\le j_1\lt \cdots\lt j_m\le n} f_{i_1\ldots i_k} g_{j_1\ldots j_m} H{\textstyle\!{\scriptstyle1\atop\scriptstyle i_1} \!{\scriptstyle\ldots\atop\scriptstyle\ldots} \!{\scriptstyle k\atop\scriptstyle i_k} \!{\scriptstyle k+1\atop\scriptstyle j_1} \!{\scriptstyle\ldots\atop\scriptstyle\ldots} \!{\scriptstyle k+m\atop\scriptstyle j_m}} .$$

This is a differential form of order $$k+m$$. It's a function in $$n+(k+m)n$$ variables.

Theorem.

$$d(\omega\land\lambda) = d\omega\land\lambda + (-1)^k\omega\land d\lambda$$

Proof: With

$$\omega = \sum_\alpha f_\alpha H_\alpha^{1\ldots k}, \qquad \lambda = \sum_\beta g_\beta H_\beta^{1\ldots m}$$

then

\eqalign{ d(\omega\land\lambda) &= \sum_{\alpha,\beta} \sum_{\nu=1}^n \left( {\partial f_\alpha\over\partial x_\nu} g_\beta + f_\beta {\partial g_\beta\over\partial x_\nu} \right) H_{\nu,\alpha,\beta}^{1\ldots k+m+1} \cr &= \sum_{\alpha,\beta} \sum_{\nu=1}^n {\partial f_\alpha\over\partial x_\nu} g_\beta H_{\nu,\alpha,\beta}^{1\ldots k+m+1} + \sum_{\alpha,\beta} \sum_{\nu=1}^n f_\alpha {\partial g_\beta\over\partial x_\nu} H_{\nu,\alpha,\beta}^{1\ldots k+m+1} \cr &= d\omega\land\lambda + (-1)^k\omega\land d\lambda,\cr }

due to

$$H_{\nu,\alpha,\beta}^{1\ldots k+m+1} = (-1)^k H_{\nu,\beta,\alpha}^{1\ldots k+m+1}$$

and

$$d\lambda = \sum_\beta \sum_{\nu=1}^n {\partial g_\beta\over\partial x_\nu} H_{\nu,\beta}^{1\ldots m+1} .$$

An alternative definition for the differentiation of $$k$$-forms could be given.

Theorem. Suppose

$$\omega = f H_\alpha^{1\ldots k}, \qquad0\le\left|\alpha\right|\le k,$$

and

$$H = \left(h_1,\ldots,h_n,h_{n+1}\right) \in R^{n\times(n+1)}$$

with $$\alpha=\left(i_1,\ldots,i_k\right)$$ we have

$$d\omega = \det\left( \mathop{\rm col}\left( \nabla f, [{\rm Id}_n]_\alpha^{1\ldots n} \right) [H]_{1\ldots n}^{1\ldots k+1} \right) = \sum_{\nu=1}^n {\partial f\over\partial x_\nu} H_{\nu,\alpha}^{1\ldots k+1},$$

where $$\rm col$$ just stacks matrices one above another and $${\rm Id}_n$$ is the identity matrix in $$R^n$$.

Proof:

$$d\omega = \left|\begin{matrix} \left\langle\nabla f,h_1\right\rangle & \ldots & \left\langle\nabla f,h_k\right\rangle & \left\langle\nabla f,h_{k+1}\right\rangle \\ \left\langle e_{i_1},h_1\right\rangle & \ldots & \left\langle e_{i_1},h_k\right\rangle & \left\langle e_{i_1},h_{k+1}\right\rangle \\ \vdots & \ddots & \vdots & \vdots\cr \left\langle e_{i_k},h_1\right\rangle & \ldots & \left\langle e_{i_k},h_k\right\rangle & \left\langle e_{i_k},h_{k+1}\right\rangle \cr \end{matrix}\right|$$
$$\qquad = \sum_{\nu=1}^n {\partial f\over\partial x_\nu} \left|\begin{matrix} h_{1,\nu} & h_{1,i_1} & \ldots & h_{1,i_k}\\ \vdots & \vdots & \ddots & \vdots\\ h_{k,\nu} & h_{k,i_1} & \ldots & h_{k,i_k}\\ h_{k+1,\nu}& h_{k+1,i_1} & \ldots & h_{k+1,i_k}\\ \end{matrix}\right|$$

since

$$\left\langle\nabla f,h_1\right\rangle = \sum_{\nu=1}^n {\partial f\over\partial x_\nu} h_{1,\nu},$$
$$\vdots\qquad\qquad\vdots$$
$$\left\langle\nabla f,h_{k+1}\right\rangle = \sum_{\nu=1}^n {\partial f\over\partial x_\nu} h_{k+1,\nu}.$$

REFERENCES.

1. Walter Rudin, Principles of Mathematical Analysis, Second Edition, McGraw-Hill, New York, 1964

2. Otto Forster, Analysis 3: Integralrechnung im $$R^n$$ mit Anwendungen, Third Edition, Friedrich Vieweg & Sohn, Braunschweig/Wiesbaden, 1984