Differential forms can be multiplied together using an operation called the wedge product. There is also a differential operator on differential forms called the exterior derivative. The wedge product of a k-form and an l-form is a (k+l)-form, and the exterior derivative of a k-form is a (k+1)-form. In particular, the exterior derivative of a 0-form (which is a function on M) is its differential (which is a 1-form on M).
The modern notion of differential forms was pioneered by Élie Cartan, and has many applications, especially in geometry, topology and physics.
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:
In particular, if v=ej 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:
Since any vector v is a linear combination ∑ vjej of its components, df is uniquely determined by dfp(ej) 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 = δij, 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
More generally, for any smooth functions gi and hi on U, we define the differential 1-form α = ∑i gi dhi pointwise by
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 fi. For n>1, such a function does not always exist: any smooth function f satisfies
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
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
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 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.
Let M be a smooth manifold. A differential form of degree k is a smooth section of the kth exterior power of the cotangent bundle of M. At any point p∈M, a k-form β defines an alternating multilinear map
The set of all differential k-forms on a manifold M is a vector space, often denoted Ωk(M).
For example, a differential 1-form α assigns to each point p∈M a linear functional αp on TpM. In the presence of an inner product on TpM (induced by a Riemannian metric on M), αp may be represented as the inner product with a tangent vector Xp. Differential 1-forms are sometimes called covariant vector fields, covector fields, or "dual vector fields", particular within physics.
The wedge product of a k-form α and an l-form β is a (k+l)-form denoted αΛβ. For example, if k=l=1, then αΛβ is the 2-form whose value at a point p is the alternating bilinear form defined by
The wedge product is bilinear: for instance, if α, β, and γ are any differential forms, then
One of the main reasons the cotangent bundle rather than the tangent bundle is used in the construction of the exterior complex is that differential forms are capable of being pulled back by smooth maps, while vector fields cannot be pushed forward by smooth maps unless the map is, say, a diffeomorphism. The existence of pullback homomorphisms in de Rham cohomology depends on the pullback of differential forms.
Differential forms can be moved from one manifold to another using a smooth map. If f : M → N is smooth and ω is a smooth k-form on N, then there is a differential form f*ω on M, called the pullback of ω, which captures the behavior of ω as seen relative to f.
To define the pullback, recall that the differential of f is a map f* : TM → TN. Fix a differential k-form ω on N. For a point p of M and tangent vectors v1, ..., vk to M at p, the pullback of ω is defined by the formula
Pullback respects all of the basic operations on forms:
The pullback of a form can also be written in coordinates. Assume that x1, ..., xm are coordinates on M, that y1, ..., yn are coordinates on N, and that these coordinate systems are related by the formulas yi = fi(x1, ..., xm) for all i. Then, locally on N, ω can be written as
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.
be a differential form and S a differentiable k-manifold over which we wish to integrate, where S has the parameterization
for u in the parameter domain D. Then [Rudin, 1976] defines the integral of the differential form over S as
is the determinant of the Jacobian. The Jacobian exists because S is differentiable.
The fundamental relationship between the exterior derivative and integration is given by the general Stokes' theorem: If is an n−1-form with compact support on M and ∂M denotes the boundary of M with its induced orientation, then
where denotes the Hodge star operator. Similar considerations describe the geometry of gauge theories in general.
The 2-form is also called Maxwell 2-form.
Numerous minimality results for complex analytic manifolds are based on the Wirtinger inequality for 2-forms. A succinct proof may be found in Herbert Federer's classic text Geometric Measure Theory. The Wirtinger inequality is also a key ingredient in Gromov's inequality for complex projective space in systolic geometry.