What is a vector filed , especially when defined generally over maniolds?

Question

In physics, I have been taught that a vector field is just assigning an arrow at each point of a manifold.

In here I read a vector field is a mapping $$ v:C^\infty(M) \rightarrow C^\infty(M)$$ So an example for a vector field would be the derivative or directional derivative. I understand the directional derivative operation takes in a smooth function and spits out another smooth function, so it makes sense with the above definition.

My question is whether the integral operator, which again takes in a smooth function and spits out another smooth function(I hope this is correct, I am not sure about the rigor of this hypothesis) is another example for a vector field in the same sense as a derivative is a vector field.

Answer 1

You seem to missing the crucial point which is that the mapping $v\colon C^\infty(M)\to C^\infty(M)$ (which ought to be really defined in $G_p^\infty(M)$: the space of germs of smooth functions at $p$)needs to satisfy the Leibniz rule: $v(fg) = f(p)v(g) + g(p) v(f)$. So an integral operator is out of question. But the idea is that a tangent vector $v\in T_pM$ is completely characterized by the values $v(f)$, where $f$ ranges over $G^\infty_p(M)$. In less "fancy" lingo, a vector is characterized by how it acts as directional derivatives.

For instance, in $\Bbb R^3$, taking a directional derivative of a function in the direction of the vector $e_1 = (1,0,0)$ is the same thing as applying $\partial/\partial x$. For this reason, the vector field that assigns to each $p\in \Bbb R^3$ the tangent vector $(1,0,0)$ in $T_p(\Bbb R^3) \cong \Bbb R^3$ is denoted by $\partial/\partial x$.

Pretty much any differential geometry book will develop these definitions accordingly. See for example Chapter 3 in Loring Tu's An Introduction to Manifolds.

Answer 2

Ivo Terek already answered the question in the last paragraph. I would like to comment on another part:

In physics, I have been taught that a vector field is just assigning an arrow at each point of a manifold.

This definition is consistent with the definition in mathematics: A vector field is a section of the tangent bundle, i.e. a vector field $X$ is a function on $M$ such that $\forall p\in M: X_p\in T_pM$. The point is that for a fixed $p\in M$ the tangent space $T_pM$ is only defined up to a unique isomorphism by its universal property. So the definition in your book is just one possible construction and you don't need to (or shouldn't) regard it as the "the only correct way" to think of tangent spaces.