Vector fields.

We have seen how to define tangent vectors at a point of a manifold. In many problems we are interested in vector fields $X$, that is a choice of vector $X(x) \in T_xM$ at every point of a manifold. We need to make sense of the notion of such a vector $X(x)$ depending smoothly on $x$. We do this as follows. Choose a chart $(U, \psi)$. Then at every point $x \in U$ we have a basis

$$\frac{\partial\phantom{\psi^1}}{\psi^1}(x), \dots, \frac{\partial\phantom{\psi^n}}{\psi^n}(x)$$

of $T_xM$ and we can expand $X(x)$ as a linear combination of these tangent vectors:

$$X(x) = \sum_{i=1}^n X^i(x) \frac{\partial\phantom{\psi^i}}{\psi^i}(x).$$

We call the functions $X^i\colon U \to \mathbb{R}$ the components of the vector field with respect to the co-ordinate chart. We have

Definition 4.5   A vector field $X$ on a manifold $M$ is smooth if its components with respect to a collection of co-ordinate charts whose domains cover $M$ are all smooth.

We have the usual lemma.

Lemma 4.4   If $X$ is a smooth vector field then its components with respect to any co-ordinate chart are smooth.

Proof. Let $(U, \psi)$ be a co-ordinate chart and let $x \in U$. Choose a co-ordinate chart $(V, \chi)$ with $x\in V$ and such that the components of $X$ are smooth with respect to $(V, \chi)$. Then write

$$X= \sum_{i=1}^n X^i(x) \frac{\partial\phantom{\psi^i}}{\psi^i}(x) =\sum_{a=1}^n X^i(x) \frac{\partial\phantom{\chi^a}}{\chi^a}(x).$$

From the results of section .. we have

$$d\chi^a = \sum_{i=1}^n \frac{\partial \chi^a }{\partial \psi^i} d\psi^i$$

so using the property of dual bases we have.

$$\frac{\partial\phantom{\psi^i}}{\psi^i} = \sum_{a=1}^n \frac{\partial \chi^a }{\partial \psi^i} \frac{\partial\phantom{\chi^a}}{\chi^a}.$$

Hence

$$X^i(y) = \sum_{a = 1}^n \frac{\partial \psi^i }{\partial \chi^a}(y) X^a(y)$$

for all $y \in U\cap V$ so that the $X^i$ are smooth on $U \cap V$ and hence smooth on all of $U$. $\qedsymbol$

Classical texts on differential geometry, in particular those on tensor calculus, downplay the co-ordinates and charts and concentrate on the components of vector fields and similar tensors. Assume that $M$ is covered by the domains of co-ordinate charts $(U_\alpha, \psi_\alpha)$. For each chart $(U_\alpha, \psi_\alpha)$ we write

$$X_{\vert U_\alpha } = \sum_{i=1}^n X^i_\alpha \frac{\partial\phantom{\psi^i}}{\psi^i}$$ (4.1)

and then as in the proof above we have that for $x$ in the intersection of $U_\alpha$ and $U_\b$ we have

$$X^i_\alpha (x) = \sum_{j = 1}^n \frac{\partial \psi_\alpha ^i }{\partial \psi_\b ^j}(x) X^j_\b (x).$$ (4.2)

The converse is also true. If we have a collection of maps $X^i_\alpha \colon U_\alpha \to \mathbb{R}$ satisfying 4.2 then we can define a vector field using 4.1 and check that it is well-defined. Classical and physics texts generally suppress the $\alpha$ index and also the sum by applying the Einstein summation convention. This convention is that any index that occurs in an expression in both a raised and lowered position is summed over. So a typical writing of 4.2 would be to say that we have co-ordinates $x^i$ and co-ordinates $x^{i'}$ and that the vector field transforms as

$$X^i = \frac{\partial x^i}{\partial x^{j'}} X^{j'}.$$

Notice that even if we do not exploit the Einstein summation convention it is a useful guide to memorising expressions like. To apply it correctly we need to remember that the index on a co-ordinate us a superscript.