Differential Form - Concept

Concept

Differential forms provide an approach to multivariable calculus that is independent of coordinates.

Let U be an open set in Rn. A differential 0-form ("zero form") is defined to be a smooth function f on U. If v is any vector in Rn, then f has a directional derivative ∂_v f, which is another function on U whose value at a point p ∈ U is the rate of change (at p) of f in the v direction:

(This notion can be extended to the case that v is a vector field on U by evaluating v at the point p in the definition.)

In particular, if v = e_j is the jth coordinate vector then ∂_vf is the partial derivative of f with respect to the jth coordinate function, i.e., ∂f / ∂xj, where x1, x2, ... xn are the coordinate functions on U. By their very definition, partial derivatives depend upon the choice of coordinates: if new coordinates y1, y2, ... yn are introduced, then

The first idea leading to differential forms is the observation that ∂_v f (p) is a linear function of v:

for any vectors v, w and any real number c. This linear map from Rn to R is denoted df_p and called the derivative of f at p. Thus df_p(v) = ∂_v f (p). The object df can be viewed as a function on U, whose value at p is not a real number, but the linear map df_p. This is just the usual Fréchet derivative — an example of a differential 1-form.

Since any vector v is a linear combination ∑ vje_j of its components, df is uniquely determined by df_p(e_j) for each j and each p∈U, which are just the partial derivatives of f on U. Thus df provides a way of encoding the partial derivatives of f. It can be decoded by noticing that the coordinates x1, x2,... xn are themselves functions on U, and so define differential 1-forms dx1, dx2, ..., dxn. Since ∂xi / ∂xj = δ_ij, the Kronecker delta function, it follows that

The meaning of this expression is given by evaluating both sides at an arbitrary point p: on the right hand side, the sum is defined "pointwise", so that

Applying both sides to e_j, the result on each side is the jth partial derivative of f at p. Since p and j were arbitrary, this proves the formula (*).

More generally, for any smooth functions g_i and h_i on U, we define the differential 1-form α = ∑_i g_i dh_i pointwise by

for each p ∈ U. Any differential 1-form arises this way, and by using (*) it follows that any differential 1-form α on U may be expressed in coordinates as

for some smooth functions f_i on U.

The second idea leading to differential forms arises from the following question: given a differential 1-form α on U, when does there exist a function f on U such that α = df? The above expansion reduces this question to the search for a function f whose partial derivatives ∂f / ∂xi are equal to n given functions f_i. For n>1, such a function does not always exist: any smooth function f satisfies

so it will be impossible to find such an f unless

for all i and j.

The skew-symmetry of the left hand side in i and j suggests introducing an antisymmetric product on differential 1-forms, the wedge product, so that these equations can be combined into a single condition

where

This is an example of a differential 2-form: the exterior derivative dα of α= ∑_j=1n f_j dxj is given by

To summarize: dα = 0 is a necessary condition for the existence of a function f with α = df.

Differential 0-forms, 1-forms, and 2-forms are special cases of differential forms. For each k, there is a space of differential k-forms, which can be expressed in terms of the coordinates as

for a collection of functions f_{i1i2 ... ik}. (Of course, as assumed below, one can restrict the sum to the case

Differential forms can be multiplied together using the wedge product, and for any differential k-form α, there is a differential (k + 1)-form dα called the exterior derivative of α.

Differential forms, the wedge product and the exterior derivative are independent of a choice of coordinates. Consequently they may be defined on any smooth manifold M. One way to do this is cover M with coordinate charts and define a differential k-form on M to be a family of differential k-forms on each chart which agree on the overlaps. However, there are more intrinsic definitions which make the independence of coordinates manifest.