## 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 [...]

Pingback by Derivatives in Coordinates « The Unapologetic Mathematician | April 6, 2011 |

[...] of the Derivative We’ve said that the tangent bundle construction is a functor with the derivative as the action on morphisms. But we haven’t actually verified that it obeys the conditions of [...]

Pingback by Functoriality of the Derivative « The Unapologetic Mathematician | April 7, 2011 |

[...] we have a canonical tangent vector in , we can hit it with the derivative and see what happens. We get a tangent [...]

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[...] Armstrong: Derivatives in Coordinates, The Derivative, Coordinate Vectors Span Tangent Spaces, Tangent Vectors at a [...]

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[...] to more general manifolds. We know that the proper generalization of the Jacobian is the derivative of a smooth map , where is an open region of an -manifold and is another -manifold. If the [...]

Pingback by The Inverse Function Theorem « The Unapologetic Mathematician | April 14, 2011 |

[...] map of manifolds is called an “immersion” if the derivative is injective at every point . Immediately we can tell that this can only happen if [...]

Pingback by Immersions and Embeddings « The Unapologetic Mathematician | April 18, 2011 |

[...] 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 [...]

Pingback by Regular and Critical Points « The Unapologetic Mathematician | April 21, 2011 |

[...] 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 [...]

Pingback by Tangent Spaces and Regular Values « The Unapologetic Mathematician | April 26, 2011 |

[...] 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 [...]

Pingback by Integral Curves and Local Flows « The Unapologetic Mathematician | May 28, 2011 |

[...] be a smooth map between manifolds, with derivative , and let and be smooth vector fields. We can compose them as and , and it makes sense to ask if [...]

Pingback by Maps Intertwining Vector Fields « The Unapologetic Mathematician | June 3, 2011 |

[...] 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 [...]

Pingback by The Adjoint Representation « The Unapologetic Mathematician | June 13, 2011 |

[...] because smooth maps push points forward. It turns out that vectors push forward as well, by the derivative. And so we can define the pullback of a -form [...]

Pingback by Pulling Back Forms « The Unapologetic Mathematician | July 13, 2011 |

[...] this pullback of we must work out how to push forward vectors from . That is, we must work out the derivative of [...]

Pingback by An Example (part 3) « The Unapologetic Mathematician | August 24, 2011 |

[...] the derivative, we see [...]

Pingback by The Tangent Space at the Boundary « The Unapologetic Mathematician | September 15, 2011 |

[...] forms are entirely made from contravariant vector fields, so we can pull back by using the derivative to push forward vectors and then [...]

Pingback by Isometries « The Unapologetic Mathematician | September 27, 2011 |

[...] The orientation on a hypersurface consists of tangent vectors which are all in the image of the derivative of the local parameterization map, which is a singular [...]

Pingback by (Hyper-)Surface Integrals « The Unapologetic Mathematician | October 27, 2011 |

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 |