71–80 of 138 posts
Stand at one of the poles. Walk to the equator, turn 90 degrees. Walk 1/4 the way around the equator, turn 90 degrees again. Then walk back to the pole. A triangle with sum 270 degrees!
Stand at one of the poles. Walk to the equator, turn 90 degrees. Walk 1/2 way around the equator, turn 90 degrees again. Walk back to the pole. Now the triangle sums 360 degrees!
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This is a tendency among physicists that I find a bit painful when reading their explanations: focusing on how things transform between coordinate systems rather than on the coordinate-independent things that are described by those coordinates. I get that these transformation properties are important for doing actual calculations, but I think they tend to obfuscate explanations. In special relativity, for example, a…
One of the worst examples is Weinberg’s book on GR, which I found nearly unreadable due to the morass of coordinates/indices. So much more painful to learn from than Wald or other mathematically modern treatments of GR.
What a terrible article. Can anyone who is not a mathematician tell me one thing they learned from this? The naked term "manifold" in its modern usage, refers to a topological manifold, loosely a locally euclidean hausdorff topological space, which has no geometry intrinsic to it at all. The hyperbolic plane and the euclidean plane are different geometries you can put on the same topological manifold, and even does n…
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Seems superficial. If a simple concept is presented in a complex way what did you actually learn?
Often, if the concept is presented in a more complex way the reason is that the author wants to emphasize and explain how the concept relates in a non-trivial way to some other deep concept; thus you learn a lot more than when the author explains things in the most simple (and shallow) way.
What a terrible article. Can anyone who is not a mathematician tell me one thing they learned from this? The naked term "manifold" in its modern usage, refers to a topological manifold, loosely a locally euclidean hausdorff topological space, which has no geometry intrinsic to it at all. The hyperbolic plane and the euclidean plane are different geometries you can put on the same topological manifold, and even does n…
This article breaks that loop and it's refreshing to see a large topic not explained as an amalgamation of arcane jargon
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> You might find it circular reasoning but it is not Um, yes it is. "A foo is an object that transforms as a foo" is a circular definition because it refers to the thing being defined in the definition. That is what "circular definition" means .
To be fair to physicists, the standard physicists' definition isn't "a tensor is a thing that transforms like a tensor", it's "a tensor is a mathematical object that transforms in the following way ". When people say "a tensor is a thing that transforms like a tensor" they're using a convenient shorthand for the bit that I put in angle brackets above. My favourite explanation is that "Tensors are the facts of the uni…
Yes, but the "convenient shorthand" only makes sense if you already know what a tensor is. That renders the "definition" useless as an explanation or as pedagogy. It's only useful as a social signal to let others know that you understand what a tensor is (or at least you think you do).
> My favourite explanation is that "Tensors are the facts of the universe"
That's not much better. "The earth revolves around the sun" is a fact of the universe, but that doesn't help me understand what a tensor is.
What matters about tensors are the properties that distinguish them from other mathematical objects, and in particular, what distinguishes them from closely related mathematical objects like vectors and arrays. Finding a cogent description of that on the internet is nearly impossible.
> the reality of the tensor ... is independent of the coordinate system chosen by the observer
Now you're getting closer, but this still misses the mark. What is "the reality of a tensor"? Tensors are mathematical objects. They don't have "reality" any more than numbers do.
> no matter how you choose your coordinates, the bases of the tensor will transform such that it "means" the same thing in your new coordinates as it did in the old ones
That is closer still. But I would go with something more like: tensors are a way to represent vectors so that the representation of a given vector is the same no matter what basis (or coordinate system) you choose for your vector space.
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It's truly the best book on Smooth Manifolds, though if you'd like a gentler approach which is still useful, then I suggest Loring Tu's books. Lee's Topological Manifolds book is also very nice. His newest edition of the Riemannian manifolds book requires selective reading or it'll slow you down.
What's the relation between the different Lee manifolds? Is it a sequence you're supposed to read in order?
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>their curvature balances out perfectly so the space does not stretch or shrink overall Could you elaborate a bit on this? I find it fascinating. Thanks. >The shape of a Calabi Yau manifold affects how particles and forces behave [...] Do you know if there's any experimental evidence of this?
> Could you elaborate a bit on this? Please correct me if I am wrong, I have not touched this subject in a long time and only have some intuition. Here is how I understand it: A manifold is a kind of space that looks flat when you zoom in close enough. The surface of a sphere or a doughnut is a 2D manifold, and the space we live in is a 3D manifold. A Calabi Yau is one of these spaces but with more dimensions and ext…
I first learned about manifolds through Introduction to Smooth Manifolds by John M. Lee. The book is dense but beautifully structured, guiding you from basic topology to smooth maps and tangent spaces with clear logic. It demands focus, yet every definition builds toward a deeper picture of how geometry works beneath the surface. Highly recommended.