The following content has been provided by the University of Erlangen-Nürnberg.

So today we come to tangent spaces to a manifold and the definitions we're going to make will be the probably single most important definitions in differential geometry,

namely that of a tangent vector to a manifold and all the other constructions we are going to derive from it.

Now, having reviewed last time the basics of vector spaces and in particular having included infinite dimensional vector spaces,

we are today going to make good use of that and the tangent space to a manifold will ultimately,

will be a finite dimensional vector space if the manifold is finite dimensional.

It will have the same dimension as a vector space as the manifold has as a topological manifold.

But first and foremost, before we come to that, we will construct an infinite dimensional vector space from a given smooth manifold

and that object will appear again and again throughout these lectures.

So, let M be a smooth manifold and actually throughout this section, throughout this chapter,

and I already emphasized that if I say M is a smooth manifold, it comes without saying that it is equipped with a topology in a smooth atlas.

I will suppress this now in the notation and only when several manifolds with several topologies and atlases are chosen

and it becomes particularly important to emphasize the differences, then I will return to writing that down explicitly.

But from now on, we are grown up, you assume this is given.

So, let M be a smooth manifold, then we can construct a vector space or the vector space over R, so over the real numbers

because we are looking at smooth real manifolds, the vector space over R whose underlying set is the set of all smooth functions on the manifold

and it is equipped with an addition and a multiplication, an S multiplication.

So, these are the smooth functions, f from M to R, but they ought to be smooth.

And then this plus is the point-wise addition.

So, whenever you have maps, you can actually define their addition.

So, say you have an f and a g, which are both smooth maps and then you define this addition, which is then an addition on C infinity of M, of course.

And you can define the resulting sum function as the sum of the functions.

This is what you mean by point-wise if you do this at every point P of the underlying manifold.

And now this here, of course, is the addition in the real numbers because f and g send a point to the real numbers.

And similarly here, you have point-wise S multiplication, which is lambda times f,

but this here is the S multiplication on this set and it is defined as lambda times f of P,

if you evaluate this at P, and this is inherited from this multiplication on R.

So, now it's my claim that this is a vector space. It's quickly checked that this is a vector space.

You believe that or you check it. It's pretty obvious.

And this is an infinite dimensional vector space, but it will play an important role.

Now, the central concept we're going to define is the following definition.

The gamma from, let's say, the whole real line into the manifold be a smooth curve.

I write from the whole range R into M because I don't want to bother about what exact range, whether it's an open interval.

Well, this is an open interval. You can always reparameterize and change that.

It's just convenient to not have to bother about this.

One could be very picky and say, well, this is just an interval from A to B,

and then you always have to worry about the domains and so on.

Actually, everything becomes very simple if we're a little bit coverly here.

And if you wanted to have it more precise, and sometimes I might want to have it more in a more precise manner,

then I will be more specific here. But let's say a curve has, for the purpose of what we're going to do here,

the full real line as a parameter range. So let gamma be a smooth curve through a point P and M.

So in a sense, we pick a point and we look at the curve through it.

Now, without loss of generality, W log, without loss of generality,

we may arrange for P to be that point where the curve is at parameter value 0.

So if it's not, you can easily define a curve with a shifted parameter value,

such that at time, if you wish, at parameter value 0, it is at the point P.

This without loss of generality, we can always arrange for this. It's just convenient to do that.

Also, this could be dropped, but it makes the formula simpler.

So now, if we have this situation, then the directional derivative operator,

that's the full name of this guy, well, not quite, then the directional derivative operator