How do I make sense of terms $X^j\partial_j(Y^i)$ in the Lie bracket of vector fields? I am trying to understand the definition of the Lie bracket of vector
fields. According to two presumably independent sources (Amari/Nagaoka
2000 and wikipedia), the Lie bracket of two vector fields $X$ and $Y$
is
$$
\begin{array}{rcl}
[X,Y]&=&\sum_{i=1}^{n}\left(X(Y^{i})-Y(X^{i})\right)\partial_{i}\\
&=&\sum_{i=1}^{n}\sum_{j=1}^{n}\left(X^{j}\partial_{j}(Y^{i})-Y^{j}\partial_{j}(X^{i})\right)\partial_{i}
\end{array}
$$
where $\partial_{i}$ is the natural basis
$\frac{\partial}{\partial\xi^{i}}$ for the tangent space at
point $p$, and $X^{i}$ as well as $Y^{i}$ are the corresponding
coordinates, i.e.
$$
X=\sum_{i=1}^{n}X^{i}\partial_{i}
$$
$$
Y=\sum_{i=1}^{n}Y^{i}\partial_{i}
$$
Here is what I do not understand. There is probably a simple answer
for this. $[X,Y]$ is a vector field, so
$\left(X(Y^{i})-Y(X^{i})\right)$ are the coordinates corresponding to
a tangent space at point $p$ of the manifold. These are supposed to be
real numbers. $X$ and $Y$ are vectors in the tangent space at point
$p$; $X^{i}$ and $Y^{i}$ are real numbers. How am I supposed to read
an expression of the form $X(Y^{i})$, a vector times a real number?
The corresponding problem in the expansion is
$X^{j}\partial_{j}(Y^{i})$ -- I am not sure how to read this
expression. $X^{j}\partial_{j}$ makes perfect sense, it's a vector in
the tangent space. But how do you multiply it by a real number? Here is the wikipedia link:
Lie bracket of vector fields
 A: For a vector field $X$ and a function $f$, it is custom to let $X(f)$ denote the derivative of $f$ in the direction $X$.
A: There is an isomorphism between vector fields $X\in\Gamma(TM)$ defined as smooth sections, i.e. $M\to TM$, and derivations $\mathrm{Der}(C^\infty(M))$. We can write this isomorphism as $X\mapsto \nabla_X$, where $\nabla_X:C^\infty(M)\to C^\infty(M)$ is such that
$$(f(p), \nabla_X f(p)) = \mathrm df\circ X(p),$$
where here $f:M\to\mathbb R$ and thus $\mathrm df:TM\to T\mathbb R\simeq \mathbb R\times\mathbb R$.
Now, when we write the expression for $X$ in coordinates, $X=X^i \partial_i$, we can interpret this expression in two different ways:

*

*as a product of functions $X^i:p\mapsto X^i(p)$ and vector fields $\partial_i$, defined so that $$\partial_i|_p\equiv \partial_i(p)\in \{p\}\times T_p M.$$

*as a product of maps $X^i$ and derivations $\partial_i\in\mathrm{Der}(C^\infty(M))$ such that
$$\partial_i f: p \mapsto \partial_i f(p)\in\mathbb R.$$
The second meaning is what people mean when writing things such as $X(f)$:
$$X(f) \equiv \eta_2\circ\mathrm df\circ X \equiv X^i \partial_i f: M\ni p\mapsto X^i(p)\partial_i f(p) \in\mathbb R,$$
where $\eta_2:T\mathbb R\to\mathbb R$ maps pairs to their second elements: $\eta_2((p,v))\equiv v$.
Things such as $Y(X(f))$ are defined similarly:
$$Y(X(f)) = Y^i \partial_i (X^j \partial_jf) = Y^i (\partial_i X^j)\partial_j f+Y^i X^j \partial_i\partial_j f.$$
A: Sometimes it is helpful to see vector fields as operators acting on functions . The action is that of a partial derivative.
With a vector field $V = V^i \partial_i$ and a function $f$:
$V(f) = V^i \frac{\partial f}{\partial_i}$
The components $V^i$ of the vector field are functions as well, so they, too, can be operated upon by another field:
$X(Y) = \left(X^j \frac{\partial Y^i}{\partial_j}\right)\partial_i$
The result is a vector field as well with components
$X(Y)^i = X^j \frac{\partial Y^i}{\partial_j}$. Being a vector field, $X(Y)$ could again act on yet another field.
Take, for example, the coordinate basis fields for polar coordinates:
$X = \partial_r = x/r \partial_x + y / r \partial_y\\
Y = \partial_\phi = -y\partial_x + x \partial_y$
The components of the fields are function of $x$ and $y$ and this is what $X$ acting on $Y$ yields:
$X(Y) = \left(\frac{x}{r}\frac{\partial(-y)}{\partial_x} + \frac{y}{r}\frac{\partial(-y)}{\partial_y}\right)\partial_x + \left(\frac{x}{r}\frac{\partial x}{\partial_x} + \frac{y}{r}\frac{\partial x}{\partial_y}\right)\partial_y = -\frac{y}{r}\partial_x + \frac{x}{r}\partial_y = \frac{1}{r}\partial_\phi$
The vector field $X(Y)$ answers the question: How does the field $Y$ change in direction of $X$?
For example, an azimuthal basis vector of polar coordinates at point $P$ changes in the direction of the local radial basis vector at $P$: Its direction is unchanged but its length changes: $X(Y) = \partial_r(\partial_\phi) = 1/r\partial_\phi$. This becomes especially clear for points $P$ on the $x$ axis: $\partial_r$ points to $+x$ direction, $\partial_\phi$ points to $+y$ direction. The vectors of $Y$ get longer farther away from the origin but don't change direction. This is just what $X(Y)$ tells you. Note that $X(Y)$ tells you what gets added to $Y$.
Hope that helps!
