Differential Forms

To begin our discussion of differential forms, let's first review the wedge product from multivariable calculus. Consider two vectors and .

The signed area of a parallelogram spanned by two vectors, and , is given by the equation , where is the acute angle between and . The formula for the area of the parallelogram yields both positive and negative answers. The area is positive if is counterclockwise from and negative if is clockwise from . Clearly the area is zero if the two vectors are parallel. Note that .

Wedge product: The wedge product of two vectors is defined as:

This is also known as the exterior product.

Consider the following diagram, where has been rotated by /2, so that the new vector is .

Then we have and algebraic expression for the wedge product:

The wedge product is analogous to the cross product in multivariable calculus, but there is one difference. When the cross product is applied to two vectors, the result is another vector. On the other hand, when the wedge product is applied to two vectors, the answer is a new expression called a 2-form, which is a type of differential form, defined below.

The following is a brief introduction to the theory of differential forms. Many of the formalities of pure mathematics have been avoided in an attempt to make the material most comprehensible.

Let's define each form and then comment on ways to interpret them.

0-form: Let K be an open set in . A 0-form is a real-valued function . When we differentiate once, it is assumed to be of class C1 , and C2 when we differentiate twice (a function is of class Cn provided its nth derivative exists and is everywhere continuous).

Basic 1-form: The basic 1-forms for the xyz-coordinate system are the expressions dx, dy, and dz. At present we consider these to be only formal symbols. A 1-form on an open set K is a formal linear combination

or just

Basic 2-form: The basic 2-forms for the xyz-coordinate system are then the expressions dx^dy, dy^dz, and dz^dx. These expressions should be thought of as products of dx and dy, dy and dz, and dz and dx. A 2-form on an open set K is a formal expression

where F, G, and H are real-valued functions on K. It is useful to note that in a 2-form, the basic 1-forms dx, dy, and dz always appear in cyclic pairs (shown in the figure below); that is, dx^dy, dy^dz, and dz^dx.

Basic 3-form: A basic 3-form for the xyz-coordinate system is a formal expression dx^dy^dz. A 3-form on an open set K is an expression of the form

where is a real-valued function on K.

Knowing these definitions does not imply that one would know how to interpret them. First, recall that a real-valued function on a domain K in is a rule that assigns a real number to each point in K; and in some sense, differential forms are generalizations of the real-valued functions studied in calculus. Note that 0-forms on an open set K are simply functions on K; that is, a 0-form f takes points in K to real numbers. Differential k-forms (k=1,2,3) can be interpreted as functions on geometric objects, such as curves and surfaces, rather than points.

Wedge product: If is a k-form and is a l-form on K, with k+l no greater than 3, the wedge product satisfies the following laws:

(i) For each k there is a zero k-form 0 with the property that for all k-forms and for all l-forms , if k+l is no greater than 3.

(ii) (Distributivity) If is a 0-form, then

.

(iii) (Anticommutativity) .

(iv) (Associativity) If are -forms, respectively, with , then

.

(v) (Homogeneity with respect to functions) If is a 0-form, then

(vi) The following multiplication rules for 1-forms hold:

(vii) If is a 0-form and is any k-form, then .

The last major step in the development of this theory is to show how to differentiate forms. The derivative of a k-form is a (k+1)-form if k<3, and the derivative of a 3-form is always zero. If is a k-form, we shall denote its derivative by . The operation has the following properties:

(1) If is a 0-form, then

.

(2) (Linearity) If and are k-forms, then

(3) If is a k-form and is an l-form,

.

(4) and .

Example 1: Let be a 1-form on some open set K in . Let's determine .

Differential forms can also be parameterized.

Finally, we have Green's theorem.

Green's Theorem: Let D be an elementary region in the xy plane, with given the counterclockwise orientation. Suppose is a 1-form on some open set K in that contains D. Then

.

Here is a 2-form on K and D is in fact a surface in parameterized by Since P and Q are explicitly not functions of z, then and so the result in example 1 reduces to . As a result, Green's Theorem can also be expressed by

References: Vector Calculus, fourth edition. Jerrold E. Marsden and Anthony J. Tromba.