differential form

1 Notation and Preliminaries.

Let M be an n-dimensional differential manifold. Let TM denote the manifold’s tangent bundle, C(M) the algebra of smooth functionsMathworldPlanetmath, and V(M) the Lie algebra of smooth vector fields. The directional derivativeMathworldPlanetmathPlanetmath makes C(M) into a V(M) module. Using local coordinates, the directional derivative operationMathworldPlanetmath can be expressed as


2 Definitions.

Differential forms.

Let A be a C(M) module. An -linear mapping α:V(M)A is said to be tensorial if it is a C(M)-homomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath, in other words, if it satisfies


for all for all vector fields vV(M) and functions fC(M). More generally, a multilinear map α:V(M)××V(M)A is called tensorial if it satisfies


for all vector fields u,,v and all functions fC(M).

We now define a differentialMathworldPlanetmath 1-form to be a tensorial linear mapping from V(M) to C(M). More generally, for k=0,1,2,, we define a differential k-form to be a tensorial multilinear, antisymmetric, mapping from V(M)××V(M) (k times) to C(M). Using slightly fancier languagePlanetmathPlanetmath, the above amounts to saying that a 1-form is a sectionPlanetmathPlanetmath of the cotangent bundle T*M=Hom(TM,), while a differential k-form as a section of Hom(ΛkTM,).

Henceforth, we let Ωk(M) denote the C(M)-module of differential k-forms. In particular, a differential 0-form is the same thing as a function. Since the tangent spacesPlanetmathPlanetmath of M are n-dimensional vector spacesMathworldPlanetmath, we also have Ωk(M)=0 for k>n. We let


denote the vector space of all differential forms. There is a natural operator, called the exterior product, that endows Ω(M) with the structureMathworldPlanetmath of a graded algebra. We describe this operation below.

Exterior and Interior Product.

Let vV(M) be a vector field and αΩk(M) a differential form. We define ιv(ω), the interior product of v and α, to be the differential k-1 form given by


The interior product of a vector field with a 0-form is defined to be zero.

Let αΩk(M) and βΩ(M) be differential forms. We define the exterior, or wedge product αβΩk+(M) to be the unique differential form such that


for all vector fields vV(M). Equivalently, we could have defined


where the sum is taken over all permutationsMathworldPlanetmath π of {1,2,,k+} such that π1<π2<πk and πk+1<<πk+, and where sgnπ=±1 according to whether π is an even or odd permutationMathworldPlanetmath.

Exterior derivative.

The exterior derivative is a first-order differential operatorMathworldPlanetmath d:Ω*(M)Ω*(M), that can be defined as the unique linear mapping satisfying

d(dα) =0,αΩk(M);
ιV(df) =v(f),vV(M),fC(M);
d(αβ) =d(α)β+(-1)kαd(β),αΩk(M),βΩ(M).

3 Local coordinates.

Let (x1,,xn) be a system of local coordinates on M, and let 1,,n denote the corresponding frame of coordinate vector fields. In other words,


where the right hand side is the usual Kronecker deltaMathworldPlanetmath symbol. By the definition of the exterior derivative,


In other words, the 1-forms dx1,,dxn form the dual coframe.

Locally, the i freely generate V(M), meaning that every vector field vV(M) has the form


where the coordinate componentsMathworldPlanetmath vi are uniquely determined as


Similarly, locally the dxi freely generate Ω1(M). This means that every one-form αΩ1(M) takes the form




More generally, locally Ωk(M) is a freely generated by the differential k-forms


Thus, a differential form αΩk(M) is given by

α =i1<<ikαi1ikdxi1dxik, (1)



Consequently, for vector fields u,v,,wV(M), we have


In terms of local coordinates and the skew-symmetrization index notation, the interior and exterior product, and the exterior derivative take the following expressions:

(ιv(α))i1ik =vjαji1ik,vV(M),αΩk+1(M); (2)
(αβ)i1ik+ =(k+k)α[i1ikβik+1ik+],αΩk(M),βΩ(M); (3)
(dα)i0i1ik =(k+1)[i0αi1ik],αΩk(M). (4)

Note that some authors prefer a different definition of the components of a differential. According to this alternate convention, a factor of k! placed before the summation sign in (1), and the leading factors are removed from (3) and (4).

Title differential form
Canonical name DifferentialForm
Date of creation 2013-03-22 12:44:46
Last modified on 2013-03-22 12:44:46
Owner rmilson (146)
Last modified by rmilson (146)
Numerical id 28
Author rmilson (146)
Entry type Definition
Classification msc 58A10
Defines exterior derivative
Defines 1-form
Defines exterior product
Defines wedge product
Defines interior product
Defines tensorial