The smooth vector fields on a manifold $M$, as a module over the ring of smooth functions on $M$

Example · Graduate

Terms

Term	This example	Why
module	✓	The smooth vector fields on a manifold $M$ form a module over the smooth functions on $M$, because the product $f X$ of a smooth function $f$ and a smooth vector field $X$ is a smooth vector field. The smooth functions form a ring but not a field, because a function that is $0$ somewhere has no multiplicative inverse.

Distinctions using this example

• Vector Space vs. Module vs. algebra

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