Differentiable Manifolds/Integral curves and Lie derivatives

Integral curves
Proof:

Let $$p \in M$$ be arbitrary, and let $$(O, \phi)$$ be contained in the atlas of $$M$$ such that $$p \in O$$.

Lemma 2.3 stated that if we denote $$\phi = (\phi_1, \ldots, \phi_d)$$, then the $$\phi_k$$, $$k \in \{1, \ldots, d\}$$ are contained in $$\mathcal C^n(M)$$. From $$\mathbf V$$ being differentiable of class $$\mathcal C^j$$ with a $$j \in \mathbb N \cup \{\infty\}$$, it follows that the functions $$\mathbf V \phi_k$$, $$k \in \{1, \ldots, d\}$$ are contained in $$\mathcal C^n(M)$$.

Thus the Picard–Lindelöf theorem is applicable, and it tells us, that each of the initial value problems
 * $$y_k'(x) = (\mathbf V \phi_k \circ y_k)(x)$$, $$k \in \{1, \ldots, d\}$$
 * $$y_k(0) = \phi_k(p)$$

has a solution $$y_k: I_k \to \mathbb R$$, where each $$I_k \subseteq \mathbb R$$ is an interval containing zero. We now choose
 * $$I := \bigcap_{k=1}^d I_k$$

and
 * $$\gamma: I \to M, \gamma(x) := \phi^{-1}(y_1(x), \ldots, y_d(x))$$

We note that
 * $$(\phi_k \circ \gamma)(x) = \phi_k(\phi^{-1}(y_1(x), \ldots, y_d(x))) = \phi(\phi^{-1}(y_1(x), \ldots, y_d(x)))_k = y_k(x)$$

Therefore we have for each $$x \in I$$ and $$k \in \{1, \ldots, d\}$$:
 * $$\gamma_x'(\phi_k) = (\phi_k \circ \gamma)'(x) = y_k'(x) = (\mathbf V \phi_k \circ \gamma)(x) = \mathbf V(\gamma(x))(\phi_k)$$

Because of theorem 2.7 then follows:
 * $$\gamma'_x = \sum_{k=1}^d \gamma_x'(\phi_k) \left( \frac{\partial}{\partial \phi_k} \right)_{\phi^{-1}(x)} = \sum_{k=1}^d \mathbf V(\phi^{-1}(x))(\phi_k) \left( \frac{\partial}{\partial \phi_k} \right)_{\phi^{-1}(x)} = \mathbf V(\gamma(x))$$

Lie derivatives
In the following, we will define so-called Lie derivatives, for
 * $$\mathcal C^n(M)$$ functions and
 * for vector fields.

So we simply defined the Lie derivative of a function in the direction of a vector field as the function defined like in definition 5.1, and the Lie derivative of a vector field in the direction of the other vector field as the lie bracket of first the first vector field and then the other (the order is important here because the Lie braket is anti-symmetric (see theorem ? and definition ?)). Since we already had symbols for these, why have we defined new symbols? The reason is that in certain circumstances, the Lie derivatives really are derivatives in the sense of limits of differential quotients, as is explained in the next chapter.