I'm taking course on differential geometry and I have a question that I haven't been able to find an answer to.
I understand that there are two equivalent definitions of a vector field on a smooth manifold $M$:
- A vector field $X: M \to TM$ is a map assigning $p\mapsto X|_p$
- A vector field $X: C^\infty(M) \to C^\infty(M)$ whereby $f\mapsto X(f) \in C^\infty(M)$ and $X(f)(p) = \sum_{i=1}^n x^i\frac{\partial f}{\partial x^i}(p)$
However, I don't understand how these definitions work on a more explict level for calculations. For example, take the vector field $X = x\frac{\partial}{\partial x} + xy\frac{\partial}{\partial y}$on $\mathbb{R}^2$, then for some $p = (x_0,y_0)\in \mathbb{R}^2$ is $X|_p$ just $x_0\frac{\partial}{\partial x} + x_0y_0\frac{\partial}{\partial y}$?
I understand that this question is quite elementary, but I haven't been able to find an explicit answer in any textbooks or articles.
Any help would be greatly appreciated.
The answer to your question is YES, but you must evaluate the partial derivatives at $p$ as well. This then gives you $X(f)(p)$ when you apply it to a smooth function.