Formulation of a vector field

Question

Let $V=y\,\partial/\partial x+x\,\partial/\partial y$ be a vector field on $\mathbb{R}^2$. I don't understand about how the vector field $V$ is defined. For example how do we map a specific point (vector) in $\mathbb{R}^2$, let say $(2,1)$, to $T(\mathbb{R}^2)$ using the given formula?

Answer 1

A vector field is a section of the tangent bundle. In other words, $$ V:\mathbb{R}^2\rightarrow T(\mathbb{R}^2) $$ such that for all $p\in \mathbb{R}^2$, $V(p)=V_p\in T_p(\mathbb{R}^2)$. In other words, for all points $p$, the vector field describes a tangent vector in the tangent space at that point.

Therefore, your $V$ is a general rule that describes how to determine a tangent vector for each point in $\mathbb{R}^2$. Let $p=(1,2)$. Then, using your definition, $$ V(p)=V_p=\left.\frac{\partial}{\partial x}\right|_p+2\left.\frac{\partial}{\partial y}\right|_p. $$ Here, you substitute the values for $p$ as the coefficients of the generators of the basis elements of the tangent space. Observe that $V(p)=V_p\in T_p(\mathbb{R}^2)$.