Differential k-form

Let $M$ be a smooth manifold and let $\pi:T^*M\to M$ be its cotangent bundle .

Definition. A differential $k$-form on $M$ is a smooth section of the vector bundle $\Lambda^k T^*M\to M$. Concretely, it is a rule that assigns to each $p\in M$ an alternating $k$-linear map

depending smoothly on $p$ (here $T_pM$ is the tangent space ).

The set of all smooth $k$-forms is denoted $\Omega^k(M)$. For $k=0$, one has $\Omega^0(M)=C^\infty(M)$. For $k=1$, a $1$-form is the same thing as a smooth covector field (a smooth section of $T^*M$), and its behavior under smooth maps is governed by the pullback of covectors (more generally by the pullback of differential forms ).

Local expression. In a smooth chart $(U,x^1,\dots,x^n)$, every $k$-form can be written uniquely as

with smooth coefficient functions $a_{i_1\cdots i_k}\in C^\infty(U)$, using the wedge product .

Two fundamental operations on forms are the wedge product $\wedge$ and the exterior derivative $d:\Omega^k(M)\to \Omega^{k+1}(M)$, which leads to the notions of closed forms , exact forms , and de Rham cohomology .

Examples

1. A $0$-form and its differential. Any smooth function $f\in C^\infty(M)$ is a $0$-form. Its exterior derivative $df$ is a $1$-form characterized by $df_p(v)=v(f)$ for $v\in T_pM$.

2. A $1$-form on $\mathbb{R}^2$. On $\mathbb{R}^2$ with coordinates $(x,y)$, the expression

   $$\omega = x\,dy - y\,dx$$

   defines a smooth $1$-form. At each point $(x,y)$, it is a covector that eats a tangent vector $(u,v)$ and returns $xv-yu$.

3. Standard volume form on $\mathbb{R}^n$. On $\mathbb{R}^n$ with coordinates $x^1,\dots,x^n$, the $n$-form

   $$dx^1\wedge\cdots\wedge dx^n$$

   is a nowhere-vanishing differential form (a smooth choice of oriented volume density).