## The Derivative

It turns out that the tangent bundle construction is actually a functor. Given a smooth map between smooth manifolds, we will get a smooth map . Yes, we’d usually write for a functor’s action on a map, but the notation is pretty classical.

So if we’re given a tangent vector we want to get a tangent vector . And since we already have sending points of to points of , it only makes sense to ask that . That is, in terms of the tangent bundle projection functions, we can write . In other words, the projection will be a natural transformation from the tangent bundle functor to the identity functor.

Anyway, this means that for each we’ll get a map . Since these are both vector spaces, it only stands to reason that we’d have a *linear* map. We haven’t yet established the connection between our “tangent vectors” and the geometric notion, but we do have a notion from multivariable calculus of a linear map that takes tangent vectors to tangent vectors: the Jacobian, which we saw as a certain extension of the notion of the derivative. We will find that our map is the analogue of the same concept on manifolds, and so we will call it the derivative of .

So here’s our definition: if is a differentiable map in some open set and if , then we define our map by

where is any smooth function on a neighborhood of . That is, is a linear functional on ; if represents a germ at we can compose it with to represent a germ at , and then we can apply itself to this germ. It should be immediately clear that this construction is linear in .

[…] take the derivative and see what it looks like in terms of coordinates. Say we have a smooth manifold and a smooth map […]

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[…] be a smooth map between manifolds. We say that a point is a “regular point” if the derivative has rank ; otherwise, we say that is a “critical point”. A point is called a […]

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[…] key observation is that the inclusion induces an inclusion of each tangent space by using the derivative . The directions in this subspace are those “tangent to” the submanifold , and so these […]

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[…] of increasing . That is, includes the interval into “at the point “, and thus its derivative carries along its tangent bundle. At each point of an (oriented) interval there’s a […]

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[…] from back to itself, and in particular it has the identity as a fixed point: . Thus the derivative sends the tangent space at back to itself: . But we know that this tangent space is canonically […]

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Hi, a quick followup please. And , again, great site; keep it up. Sorry for the ASCII; ignore it if it is too hard,sorry don’t know how to do otherwise.

We’re given a map F:M–>N between manifolds.

So,please , let me see if I understand f_*p well, using the curve def. of tangent vector v based at p.

Consider a chart (U,Phi) containing p in M:, and a curve:

C(t):(-1,1)–>M ; C(0)=p we define a tangent vector v in TpM in terms of the derivative (living in R^n) given by: d/dt( (Phi)oC)(0), so that

.

C(t) represents a tangent vector v_p in TpM (specifically, v_p representis all curves C_i with the same number value d/dt(PhioC_i)(0) ) .

Then F: M–>N sends this vector/curve-class v_p in M to the curve FoC in N. Now, this curve is itself

a tangent vector based at F(p) in N; we consider a chart (W, Csi) containing F(p). Then F_* (v_p) is the class given by

d/dt(Csi oFo C)(0).

Is that It?

Thanks again for the nice site. Let me know if you write a book ( I guess it would be an E-book).

Comment by carl | May 30, 2013 |

Yes, Carl, that’s it. Geometrically we just watch how the map takes a curve passing through and maps it to a curve passing through , and we observe that the tangent vector of the target curve at is independent of everything but the tangent vector of the original curve at .

Comment by John Armstrong | May 30, 2013 |

Thanks, John:

But, in practice, say we have the usual setup: C:(-1,1)–>M c(0)=p and (U,Phi) a chart containing p. Then we consider the curve

C':=Phi oC landing in R^n. Do we actually consider C’ as a curve in R^n, or do we transplant/pullback C’ from R^n into T_pM by the

chart maps?

Comment by carl | May 30, 2013 |

Well, some of this seems a little confused; I’m really not sure what you’re trying to do here.

Comment by John Armstrong | May 30, 2013 |

O.K, I thought I was using the tangent plane as a sub for R^n ; the curve in R^n could live instead in

the tangent space, which is a copy of R^n. But maybe I am confused.

Comment by carl | May 30, 2013 |

Yes, the curve does not live in the tangent space. A curve is a function from a parameter interval to a manifold.

Comment by John Armstrong | May 30, 2013 |

I see; thanks again, I’ll go back to the books to clear things up.

Comment by carl | May 30, 2013 |