**Abstract.** This article will give a very simple definition of -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 -forms vanishes.

**MSC 2010:** 58A10

# 1. Basic definitions.

We denote the submatrix of consisting of the rows and the columns with

and its determinant with

For example

Suppose

and let

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

a *basic -form* or *basic differential form* of order . It’s a real function of variables. For the expression is defined to be zero. If also depends on then

is called a *-form*. It’s a real function of variables which is -linear in the column-vectors of .

For example for and we have . This is a linear function in and a possibly non-linear function in .

# 2. Differentiation of -forms.

For the differential form

we define

as the *outer differentiation* of . This is a -form. It’s a function of variables.

The -form

yields

which corresponds to .

In the special case we get for

the result

This corresponds to .

Let hat () mean exclusion from the index list. The case for

delivers

This corresponds to .

**Theorem.** For we have

*Proof:* With

we get

and this is zero, because

and

Application of this theorem to an -form with an and a -form with an reading (1) and then (2) yields

The second equation is only true for because

**Definition.** Suppose

is differentiable, its derivative denoted by , and

For the differential form we define the *back-transportation* as

and the integral over -forms as

For example the case ,

gives

# 3. The outer product of differential forms.

Suppose

For the two differential forms

and

the outer product is defined as

This is a differential form of order . It’s a function in variables.

**Theorem.**

*Proof:* With

then

due to

and

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

**Theorem.** Suppose

and

with we have

where just stacks matrices one above another and is the identity matrix in .

*Proof:*

since

REFERENCES.

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

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