Show that locally a differential form $\omega$ equals $\Sigma _I\omega _Idx^I$ in which $\omega _I$ are smooth functions

Question

First I give the definition of differential form that I'm using.

Definition: Let $M\subseteq\mathbb{R}^n$ be a smooth submanifold and $p\in\mathbb{N}^\times $. We say that $\omega:M\to \cup _{x\in M}\Lambda ^p(T_xM)^* $ is differential $p$-form (which we denote by $\omega \in \Omega ^p(M)$) if the following propositions are true:

1. $\omega (x)\in \Lambda ^p(T_xM)^*$ for all $x\in M$;
2. $M\to \mathbb{R},\,x\mapsto \omega (x)(X_1(x),\cdots,X_p(x))$ is smooth for all smooth vector fields $X_1,\cdots,X_p:M\to\mathbb{R}^n$.

My question is: Suppose that $\omega \in \Omega ^p(M)$. How can I show that locally $\omega $ equals $\Sigma _I\omega _Idx^I$ in which $\omega _I$ are smooth functions?

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I already proved that locally $\omega $ equals $\Sigma _I\omega _Idx^I$ in which $\omega _I$ are functions, however I don't know how to prove that those functions are smooth. My problem is with the following fact:

• Suppose that $A\subseteq M$ is open in $M$. I don't know how to prove that the restriction $\omega |_A$ is also a differential $p$-form. It's easy to prove that $\omega (x)\in \Lambda ^p(T_xA)^*$ for all $x\in A$ but I don't know how to prove that $A\to \mathbb{R},\,x\mapsto \omega (x)(X_1(x),\cdots,X_p(x))$ is smooth for all smooth vector fields $X_1,\cdots,X_p:\color{red}{A}\to \mathbb{R}^n$ because we can't guarantee that given a smooth vector field $X:A\to \mathbb{R}^n$ in $A$ there's a smooth vector field $\tilde X:M\to \mathbb{R}^n$ in $M$ such that $\tilde X|_A=X$. I tried to solve this problem using partition of unity and coordinate vector fields $\frac{\partial}{\partial x_i}$ but I failed.

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Please don't assume that $\cup _{x\in M}\Lambda ^p(T_xM)^*$ is a smooth manifold because I'm only studying submanifolds of $\mathbb{R}^n$ and I'm not considering the general definition of manifold.

Answer 1

You can use smooth bump functions. Say $A$ is the coordinate patch you are interested in, then for any $x\in A$ consider a small closed disk $D\subset A$ centered at $x$. There then exists a bump function $b\colon M\to\mathbb{R}$ with support in $A$ such that $b|_D\equiv 1$ is constant. Then consider the global vector fields $X_i := b\cdot \partial_i$, and using property (2) is is easy to show that the the functions $b\cdot\omega_i$ then are smooth. We conclude that then $\omega_i$ is smooth at $x$, and since this can be done for any point in $A$ we have shown the $\omega_i$ are smooth.

Answer 2

The idea is to take vector fields $ X_1,… X_p $ such that $ \omega(X_1,… X_p) = \omega_I$. Taking $A$ to be a coordinate open set, if $I=i_1i_2…i_p$, take $ \frac{\partial}{\partial x_{i_1}},…, \frac{\partial}{\partial x_{i_p}}$ to be the vector field. And the condition $2)$ says that $ \omega_I = \omega(\frac{\partial}{\partial x_{i_1}},…, \frac{\partial}{\partial x_{i_p}}) $ is smooth.