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:and multiplied by smooth functions on M

:

to get other vector fields. The set of all vector fields then takes on the structure of a module over the commutative algebra of smooth functions on M, denoted .

A local vector field on is a local section of the tangent bundle. That is, a local vector field is defined only on some open set and assigns to each point of a vector in the associated tangent space. The set of local

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