What is a Manifold? Lesson 10: Tangent Space - Basis Vectors

What is a Manifold? Lesson 10: Tangent Space - Basis Vectors

[LadWithADog]

What an astounding lecture, clearest I have ever been seen on the subject

[javadrazavi]

beautiful, clear and enjoyable representation of the subject

[RahulMadhavan]

Thanks for all the geometric interpretation! Having gone through all that, the algebra makes sense now.

[febojarlock]

Eres excelente explicando! muy buen maestro!!

[StudentOfKyoto]

Really love the fact that you are pointing out all the implicit renaming to make it extra clear.

I believe at 50:50 you wanted to say X_\phi^j = X_\gamma^i dy^j/dx^i . But the meaning is clear from the context - that they follow "usual" coordinate transformation.

Thanks a ton!

[tursinbayoteev]

At 33:15, this very similar to the definition of the basis vectors of the covariant vectors.

[nafriavijay]

So basis vectors are velocities for a test function ?... I hope you are getting what I mean. In a sense that on a manifold they tells you how a curve (from a real line to manifold ) are transitioning wrt t_0, at what rate. On evaluation with a function f (Test function) at a point p on manifold with a t_0 we get the "Real Number". For every dimension. Just like in Tensors lecture.

[olehvinichenko]

then, do these basis vectors/covectors for tangent/cotangent spaces can construct a tensor product space? switch between abstract notion of tensor as an object in TPS as multi-linear map and the one from differential geometry construction failing on me, sorry about silly question

[Minus__form_symmetry]

very good lectures.i would request you to extend the series to levi civita connections etc.(from a perspective of a physics student that would very useful)

[chengmingliu]

Thanks for the lucture! It is nice and clear.


May I ask that If we map the curve corresponding to a basis vector to the chart's euclidean space R^d, are they parallel to the axis in R^d?


And when we want to express a curve using the basis, we just map the curve to the euclidean space R^d, and see how does the coordinate change with respect to the time t, and it suggest the indexes of the linear combination?

[albertocarraro]

What book would match best this set of lectures, in your opinion, to serve as a reference?

[hannesstark]

Yeah man I really think you should write those sum symbols sometimes :D

[ThomasImpelluso]

Also, at 10:00, you deploy your test function, f. And it goes from the manifold to the real numbers. But, please correct me if I am wrong...

f, being a diffeomorphism, implies, to me, that the region, R, to which "f" maps is ALSO a manifold.

And for any of this to make sense, you SHOULD map that region R, to another chart, although an R1 chart.

HOWEVER, this mapping can be modeled as the Identy mapping, so you just casually, get a bit lazy and do your integration on the "manifold" to which f maps.

[petershotts]

Thank you so much for these lectures. I have just about managed to hang on whilst going through the last two lectures, but the gain from the endurance has been huge! I have a very general two part question. I am struck by the need throughout to use intervening topological spaces with the standard Euclidean topology in order to establish the concept of differentiability. Does this mean that it is impossible to establish a concept of differentiability without reference to the standard Euclidean topology? Alternatively can one arrive at results in which a satisfactory concept of differentiation is defined through the intervention of an alternative topology - e.g. hyperbolic geometry? Clearly if the answer is yes, but the conclusions reached are the same then there is little to be gained from the inconvenience of working with an alternative intervening topology. The second part to my question, therefore is, if it is possible to choose an alternative intervening topology can this lead to different sets of results?

[theodorei.]

If I get this right, the coefficients a' of a vector in a tangent space are independent of the test function that you will use, if the chart, point and the curve a are fixed, correct?

[acousticsoundwave]

At 22:43, am I right in thinking that the chart-dependent term for the velocity is an einstein sum of those partials? So that chain rule term is actually \sum_i^d(complicated chain rule)?

[rasraster]

Love your lectures. But I must say that your choice of "x^i" for the vector components while using "x_i" for the coordinates in γ is *really* problematic for me. Thanks for doing this series, it's a treasure.

[howieg]

Would have been a LOT clearer in change of basis, if you had used the mapping X and Y and not got involved with all the Greek letters.
Since everything is already overloaded in DiffGeom . I mean Chart X does mean the mapping is X and for most authors x1, x2, ... are the components.

[ThomasImpelluso]

Could I ask you. You whimsically said that a velocity vector is just something that measures how a function changes as we pass along a path on a manifold.

So, in that sense, the "velocity vector" is a mis-statement: perhaps we should call it a "change vector" or something like that.

For, if you really want to call it a velocity, then I have a problem. A velocity MEANS something to me (it is speed with a direction). So, in that sense, is there a PREFRRED function, f, that is used to reveal a "velocity" along a curve as its home-mapping on R-1 (time) changes?

In other words, what is the form of "f" to REALLY make this thing a "velocity" in the traditional sense?

[ThomasImpelluso]

At 10:01, why did you get lazy and say that "f" went from the manifold to the Reals. I would have preferred you also added that it went from the manifold to another manifold, as that is how I got comfortable with it. You seem to be abandoning the fact that THAT R is a manifold. Should I relax?

For when you get lazy like this (no offense intended), then I am forced to wonder why you could not have taken the deriviative on that first Real line where "t" is the parameter.

In other words, I see the curve as parameterized by "t": thus, that first real line from which alpha draws. But I cannot now distinguish why you need the second real line. Could you referesh my memory before I go back?