differential form

A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors.

Gentle introduction

We initially work in an open set in Rⁿ. A 0-form is defined to be a smooth function f. When we integrate a function f over an m-dimensional subspace S of Rⁿ, we write it as

\int_S f\,dx_1 \ldots dx_m.

Consider dx₁, ..., dx_n for a moment as formal objects themselves, rather than tags appended to make integrals look like Riemann sums. We call these and their negatives −dx₁, ..., −dx_n basic 1-forms. We define a "multiplication" rule ∧, the wedge product on these elements, making only the anticommutativity restraint that

dx_i \wedge dx_j = - dx_j \wedge dx_i

for all i and j. Note that this implies

dx_i \wedge dx_i = 0

We define the set of all these products to be basic 2-forms, and similarly we define the set of products

dx_i \wedge dx_j \wedge dx_k

to be basic 3-forms, assuming n is at least 3. Now define a monomial k-form to be a 0-form times a basic k-form for all k, and finally define a k-form to be a sum of monomial k-forms. We extend the wedge product to these sums by defining

(f\,dx_I + g\,dx_J)\wedge(p\,dx_K + q\,dx_L) =

f \cdot p\,dx_I \wedge dx_K +

f \cdot q\,dx_I \wedge dx_L + g \cdot p\,dx_J \wedge dx_K + g \cdot q\,dx_J \wedge dx_L, etc., where dx_I and friends represent basic k-forms. In other words, the product of sums is the sum of all possible products. Now, we also want to define k-forms on smooth manifolds. To this end, suppose we have an open coordinate cover. We can define a k-form on each coordinate neighborhood; a global k-form is then a set of k-forms on the coordinate neighborhoods such that they agree on the overlaps. For a more precise definition what that means, see manifold.

Properties of the wedge product

It can be proved that if f, g, and w are any differential forms, then

w \wedge (f + g) = w \wedge f + w \wedge g.

Also, if f is a k-form and g is an l-form, then:

f \wedge g = (-1)^{kl} g \wedge f.

Hardcore (but brief) definition and discussion

In differential geometry, a differential form of degree k is a smooth section of the kth exterior power of the cotangent bundle of a manifold. At any point p on a manifold, a k-form gives a multilinear map from the k-th cartesian power of the tangent space at p to R. For example, the differential of a smooth function on a manifold (a 0-form) is a 1-form. 1-forms are a particularly useful basic concept in the coordinate-free treatment of tensors. In this context, they can be defined as real-valued linear functions of vectors, and they can be seen to create a dual space with regard to the vector space of the vectors they are defined over. An older name for 1-forms in this context is "covariant vectors".

Integration of forms

Differential forms of degree k are integrated over k dimensional chains. If k = 0, this is just evaluation of functions at points. Other values of k = 1, 2, 3, ... correspond to line integrals, surface integrals, volume integrals etc. Let

\omega=\sum a_{i_1,\cdots,i_k}({\mathbf x})dx_{i_1} \wedge \cdots \wedge dx_{i_k}

be a differential form and S a set for which we wish to integrate over, where S has the parameterization

S({\mathbf u})=(x_1({\mathbf u}),\cdots,x_n({\mathbf u}))

for u in the parameter domain D. Then 1976 defines the integral of the differential form over S as

\int_S \omega =\int_D \sum a_{i_1,\cdots,i_k}(S({\mathbf u})) \frac{\partial(x_{i_1},\cdots,x_{i_k})}{\partial(u_{1},\cdots,u_{k})}d{\mathbf u}

where

\frac{\partial(x_{i_1},\cdots,x_{i_k})}{\partial(u_{1},\cdots,u_{k})}

is the determinant of the Jacobian. See also Stokes' theorem.

Operations on forms

The set of all k-forms on a manifold is a vector space. Furthermore, there are two other operations: wedge product and exterior derivative. d² = 0, see de Rham cohomology for more details. The fundamental relationship between the exterior derivative and integration is given by the general Stokes' theorem, which also provides the duality between de Rham cohomology and the homology of chains.

References

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