Identifying Vector Fields
We know what vector fields are on a region , but to identify them in the wild we need to verify that a given function sending each to a vector in is smooth. This might not always be so easy to check directly, so we need some equivalent conditions. First we need to define how vector fields act on functions.
If is a vector field and is a smooth function then we get another function by defining . Indeed, , so it can take (the germ of) a smooth function at and give us a number. Essentially, at each point the vector field defines a displacement, and we ask how the function changes along this displacement. This action is key to our conditions, and to how we will actually use vector fields.
Firstly, if is a vector field — a differentiable function — and if is a chart with , then is always smooth. Indeed, remember that gives us a coordinate patch on the tangent bundle. Since is smooth and is smooth, the composition
is also smooth. And thus each component is smooth on .
Next, we do not assume that is a vector field — it is a function but not necessarily a differentiable one — but we assume that it satisfies the conclusion of the preceding paragraph. That is, for every chart with each is smooth. Now we will show that is smooth for every smooth , not just those that arise as coordinate functions. To see this, we use the decomposition of into coordinate vector fields:
which didn’t assume that was smooth, except to show that the coefficient functions were smooth. We can now calculate that , since
But this means we can write
which makes a linear combination of the smooth (by assumption) functions with the coefficients , proving that it is itself smooth.
Okay, now I say that if is smooth for every smooth function on some region , then is smooth as a function, and thus is a vector field. In this case around any we can find some coordinate patch . Now we go back up to the composition above:
Everything in sight on the right is smooth, and so the left is also smooth. But this is exactly what we need to check when we’re using the local coordinates and to verify the smoothness of at .
The upshot is that when we want to verify that a function really is a smooth vector field, we take an arbitrary smooth “test function” and feed it into . If the result is always smooth, then is smooth. In fact, some authors take this as the definition, regarding the action of on functions as fundamental, and only later talking in terms of its “value at a point”.