Lecture 5: Tangent Spaces (International Winter School on Gravity and Light 2015)
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- čas přidán 9. 02. 2015
- As part of the world-wide celebrations of the 100th anniversary of Einstein's theory of general relativity and the International Year of Light 2015:
Central lecture course by Frederic P Schuller (A thorough introduction to the theory of general relativity) introducing the mathematical and physical foundations of the theory in 24 self-contained lectures at the International Winter School on Gravity and Light in Linz/Austria.
The lectures develop the theory from first principles and aim at an audience ranging from ambitious undergraduate students to beginning PhD students in mathematics and physics.
Satellite Lectures (see other videos on this channel) by Bernard F Schutz (Gravitational Waves), Domenico Giulini (Canonical Formulation of Gravity), Marcus C Werner (Gravitational Lensing) and Valeria Pettorino (Cosmic Microwave Background) expand on the topics of this central lecture course and take students to the research frontier.
Access to further material on www.gravity-and-light.org/lectures and www.gravity-and-light.org/tutorials
"Vectors in differential geometry survive as the directional derivatives they induce."
a miraculous gift to general public passionate for this subject.The exposition is beyon exelence miguel Caracas Venezuela
Bless that woman and her questions :)
He should have fed an f at the far left and right of the equation at 22:52
First I Thank you for these precious lectures professor and I thank the school for uploading the videos on youtube, it worth a million dollar. But I just don't get the final part of the demonstration from 32:32 , could someone explain?
It's been 9 months, so you've probably figured this out by now, but I'll explain anyways.
Since there is a composition of functions (f o g), and g(t) is multi-valued, it's simplest to talk about the derivative of one of the components of g(t) at a time, hence g(t)[i]', where g(t)[i] is the i-th components of g(t).
Since f maps from R^d to R, f's derivative depends on all components of its argument g(t). In fact, we know from the multivariate chain rule that
f'(g(t)) = sum from i=1 to i=d { ( d/dg(t)[i] f(g(t)) ) * g(t)[i]' }
In the video, Frederic uses implicit summation via Einstein notation for this. Remember, d/dg(t)[i] is a partial derivative OPERATOR, and g(t)[i]' is the derivative of the i-th component of g w.r.t. the parameter t, evaluated at t.
In the last step of the proof, he distributed the multiplication, resulting in two terms, and reverses the chain rule on each term separately.
Did anyone attend this and still have the questions from the tutorials?
Maybe you refer to these ones: gravity-and-light.herokuapp.com/tutorials
you're amazing!
I wonder If someone can help because I am losing sleep over the corollary at 1:07:37. We have a basis for TpU, but why does the dimension, d, of this basis necessarily imply that the dimension of TpM and M is also d? The chart map sends M->R^d so it seems the dimension could change for different charts. The only thing I can think of is that charts with "redundant" dimensions would not have a linearly independent basis. I'm also puzzled by the manifold dimension. I thought dimension was only discussed for vector spaces that have a notion of addition and multiplication. A topological manifold does not have these propertied so how could we talk about its dimension? Thanks.
I'm pretty sure the idea is as follows: the dimension on a manifold M is the defined to be the dimension of the Euclidean space R^d that the charts map into. Completely unrelated a priori, the dimension of T_pM is the size of a basis of this space. Now, the point p in M is contained in a chart, so you can look at the image of this chart in R^d. Rig it up so that the point p is mapped to the origin. Then, the standard basis d/dx1, ..., d/dxn of R^d, essentially define for you a basis of TpM. Thus, the two dimensions, a priori unrelated, actually agree. This argument might not be perfect, but it's the idea!
At every point of the manifold you have an homeomorphism (this is highly important! not just any map) from an open subset U of M to an open subset V of R^d. You define the dimension of the manifold as this d. The fact that this is actually well defined - i.e. you cant have two charts to R^d and R^e with d != e - is highly non trivial. The point is that the chart transition map would yield a homeomorphism of an open subset of R^d to an open subset of R^e, but this implies d = e by Brouwer's theorem of invariance of dimension (which is the non trivial part).
For differentiable manifolds, however, this proof becomes trivial. Because the chart transition then is a diffeomorphism (not just homeomorphism) which implies (by "undergraduate differentiation") that its differential needs to be invertible. But its differential is a linear map between vector spaces of dimension d and e so it can only be invertivle if d=e.