# Differentiable Manifolds/Integral curves and Lie derivatives

## Integral curves[edit | edit source]

**Definition 8.1**:

Let be a manifold, let , and let be an interval. An **integral curve for ** is a function such that

**Theorem 8.2**:

Let be a manifold of class , (it is important that here), let be differentiable of class , and let . Then there exists an interval and an integral curve of such that and .

**Proof**:

Let be arbitrary, and let be contained in the atlas of such that .

Lemma 2.3 stated that if we denote , then the , are contained in . From being differentiable of class with a , it follows that the functions , are contained in .

Thus the Picard–Lindelöf theorem is applicable, and it tells us, that each of the initial value problems

- ,

has a solution , where each is an interval containing zero. We now choose

and

We note that

Therefore we have for each and :

Because of theorem 2.7 then follows:

## Lie derivatives[edit | edit source]

In the following, we will define so-called *Lie derivatives*, for

- functions and
- for vector fields.

**Definition 8.3**:

Let be a manifold of class , and . The **Lie derivative of in the direction of **, denoted by , is defined as follows:

**Definition 8.4**:

Let be a manifold and . The **Lie derivative of in the direction of **, denoted by , is defined as follows:

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.