differential form

Notation and Preliminaries.

Let $M$ be an $n$ -dimensional differential manifold. Let $TM$ denote the manifold's tangent bundle, $C^\infty(M)$ the algebra of smooth functions, and $ V(M)$ the Lie algebra of smooth vector fields. The directional derivative makes $C^\infty(M)$ into a $V(M)$ module. Using local coordinates, the directional derivative operation can be expressed as$$ v(f) = v^i \partial_i f,\quad v\in V(M),\; f\in C^\infty(M)$$

Definitions.

Differential forms.

Let $A$ be a $C^\infty(M)$ module. An $ \mathbb{R}$ -linear mapping $ \alpha: V(M)\to A$ is said to be tensorial if it is a $C^\infty(M)$ -homomorphism, in other words, if it satisfies$$ \alpha(f v) = f \alpha(v) $$ for all for all vector fields $v\in V(M)$ and functions $f\in C^\infty(M)$ . More generally, a multilinear map $\alpha: V(M)\times\dots\times V(M)\to A$ is called tensorial if it satisfies$$ \alpha(f u,\dots,v) = \cdots = \alpha(u,\dots, fv) = f \alpha(u,\dots, v)$$ for all vector fields $u,\dots,v$ and all functions $f\in C^\infty(M)$ .

We now define a differential 1-form to be a tensorial linear mapping from $V(M)$ to $C^\infty(M)$ . More generally, for $k=0,1,2,\ldots, $ we define a differential $k$ -form to be a tensorial multilinear, antisymmetric, mapping from $V(M)\times \cdots \times V(M)$ ($k$ times) to $C^\infty(M)$ . Using slightly fancier language, the above amounts to saying that a $1$ -form is a section of the cotangent bundle $T^*M = \Hom(TM,\Rset)$ , while a differential $k$ -form as a section of $\Hom(\Lambda^k TM,\Rset)$ .

Henceforth, we let $\Omega^k(M)$ denote the $C^\infty(M)$ -module of differential $k$ -forms. In particular, a differential $0$ -form is the same thing as a function. Since the tangent spaces of $M$ are $n$ -dimensional vector spaces, we also have $\Omega^k(M)=0$ for $k>n$ . We let$$\Omega(M) = \bigoplus_{k=0}^n \Omega^k(M$$ denote the vector space of all differential forms. There is a natural operator, called the exterior product, that endows $\Omega(M)$ with the structure of a graded algebra. We describe this operation below.

Exterior and Interior Product.

Let $v\in V(M)$ be a vector field and $\alpha\in \Omega^k(M)$ a differential form. We define $\iota_v(\omega)$ , the interior product of $v$ and $\alpha$ , to be the differential $k-1$ form given by$$ \iota_v(\alpha)(u_1,\dots,u_{k-1}) = \alpha(v,v_1,\dots,v_{k-1}),\quad v_1,\dots,v_{k-1}\in V(M)$$ The interior product of a vector field with a $0$ -form is defined to be zero.

Let $\alpha\in \Omega^k(M)$ and $\beta\in\Omega^\ell(M)$ be differential forms. We define the exterior, or wedge product $\alpha\wedge\beta\in\Omega^{k+\ell}(M)$ to be the unique differential form such that$$ \iota_v(\alpha\wedge\beta) = \iota_v(\alpha)\wedge \beta + (-1)^k \alpha \wedge \iota_v(\beta$$ for all vector fields $v\in V(M)$ . Equivalently, we could have defined$$ (\alpha\wedge\beta)(v_1,\dots, v_{k+\ell}) = \sum_{\pi}\operatorname{sgn}(\pi) \alpha(v_{\pi_1},\dots,v_{\pi_k}) \beta(v_{\pi_{k+1}},\dots,v_{\pi_{k+\ell}})$$ where the sum is taken over all permutations $\pi$ of $\{1,2,\dots, k+\ell\}$ such that $\pi_1< \pi_2 < \cdots \pi_k$ and $\pi_{k+1} < \cdots < \pi_{k+\ell}$ , and where $\operatorname{sgn} \pi=\pm 1$ according to whether $\pi$ is an even or odd permutation.

Exterior derivative.

The exterior derivative is a first-order differential operator $d:\Omega^*(M)\rightarrow \Omega^*(M)$ , that can be defined as the unique linear mapping satisfying

Local coordinates.

Let $(x^1,\ldots,x^n)$ be a system of local coordinates on $M$ , and let $\partial_1,\dots,\partial_n$ denote the corresponding frame of coordinate vector fields. In other words,$$ \partial_i(x^j) = \delta_i{}^j$$ where the right hand side is the usual Kronecker delta symbol. By the definition of the exterior derivative,$$\iota_{\partial_i} (dx^j) = \delta_i{}^j$$ In other words, the 1-forms $dx^1,\dots,dx^n$ form the dual coframe.

Locally, the $\partial_i$ freely generate $V(M)$ , meaning that every vector field $v\in V(M)$ has the form$$ v= v^i \partial_i,$$ where the coordinate components $v^i$ are uniquely determined as$$ v^i=v(x^i).$$ Similarly, locally the $dx^i$ freely generate $\Omega^1(M)$ . This means that every one-form $\alpha\in\Omega^1(M)$ takes the form$$\alpha=\alpha_i dx^i$$ where$$\alpha_i=\iota_{\partial_i} (\alpha)$$ More generally, locally $\Omega^k(M)$ is a freely generated by the differential $k$ -forms$$dx^{i_1}\wedge\cdots\wedge dx^{i_k},\qquad 1\leq i_1<i_2<\cdots<i_k\leq n$$ Thus, a differential form $\alpha\in \Omega^k(M)$ is given by

where$$ \alpha_{i_1\dots i_k} = \alpha(\partial_{i_1},\dots, \partial_{i_k})$$ Consequently, for vector fields $u,v,\dots, w\in V(M)$ , we have$$ \alpha(u,v,\dots,w) = \alpha_{i_1i_2\dots i_k} u^{i_1} v^{i_2}\cdots w^{i_k}$$

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

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).