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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
Consider dx1, ..., dxn 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, ..., −dxn basic 1-forms.
We define a "multiplication" rule ∧, the wedge product on these elements, making only the anticommutativity restraint that
for all i and j. Note that this implies
We define the set of all these products to be basic 2-forms, and similarly we define the set of products
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
etc., where dxI 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.
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".
Differential forms of degree k are integrated over k dimensional chainIn set theory, a chain is a total order subset of a poset. See also Ascending Chain Condition and Descending Chain Condition. Algebraic topology In algebraic topology, a simplicial k chain is a formal linear combination of k simplices. Integration on chais. 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.
See also Stokes' theoremStokes' theorem in differential geometry is a statement about the integration of differential forms which generalizes several theorems from vector calculus. It is named after Sir George Gabriel Stokes ( 1819- 1903). Let M be an oriented piecewise smooth m.