We can take the space of vector fields on a smooth manifold .
This is a vector space, since elements can be added and scaled pointwise
In this way, it uses the vector space structures of the fibers.
As a Lie algebra
For a vector field and a smooth function , we can combine these in several ways.
We can multiply the function and the vector field
We can also evaluate on ( is an operator on the smooth functions .)
If is defined on an open set ,
Thus, for a function , we can view as its own smooth function, and we can apply another vector field to it,
The only problem is that this new object is not guaranteed to be a derivation, so while it is an operator on smooth functions (i.e. ) it may not be in the tangent space.
So, instead we can use the special commutator bracket where