Two differential forms are the same if they are the same locally.

Question

Two differential forms are the same if they are the same locally.

Why is it the case? Could anyone give me some suggestions in this regard?

Thanks for your time.

Answer 1

A differential $k$-form on $M$ is a section of the $k$th exterior power of the cotangent bundle... Whatever that means.

Important for your question is that a differential $k$-form is a special smooth function $\alpha : M \to \bigwedge^k T^* M$, where the codomain is just some manifold (which was cleverly constructed to classify differential $k$-forms on $M$!). So your question is really about whether two smooth functions which agree locally must be the same.

If by "locally" you mean "for each point $p$ there's a neighborhood $U$ so that $\alpha \restriction_U = \beta \restriction_U$", then yes -- $\alpha = \beta$, since they're just functions.

If by "locally" you mean that there's some neighborhood where $\alpha \restriction_U = \beta \restriction_U$, then not necessarily. After all, there's no reason two smooth functions which agree in some neighborhood have to agree everywhere!

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As an aside, this is one of the reasons to gain some familiarity with these high-abstraction ways of talking about $k$-forms (or other mathematical topics too). For computations, you almost always want to be working with coordinate charts, but for these kinds of "does everything work nicely?" abstract questions, having some machinery with a simpler language can help make certain things obvious that are far from obvious when we have to worry about coordinate charts and transition functions.

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I hope this helps ^_^