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You're right, there is room for improvement in this article. Something that isn't made perfectly clear is that these tangent spaces, as described in this article, are treated as abstract manifolds. For your sphere embedded in R^3, the tangent planes are not considered as subsets of the same R^3,
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but as vector spaces that have an independent existence. So the tangent bundle of a two-dimensional sphere doesn't live inside 3-space; it really is a four dimensional manifold (a two-dimensional family of two-dimensional planes; two plus two makes four).
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Question, isn’t this a derivative at a point instead of a derivation? Sorry to be pedantic but from what I understand of derivation, the codomain here isn’t the family of smooth functions and the reals aren’t a bimodule over the ring of smooth functions.
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Everything seems to render correctly for me. Sometimes LaTeX formulas, which render as PNG graphics, don't get cached right away if the server load is high, and that might be the problem. At any rate, give it a few hours, and it should be ok.
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Okay, a sphere has dimension two and is embedded in 3-space. Every tangent space of that sphere is a plane. Combining them all gives you the universe, minus the inside of the sphere, and is called the tangent bundle. It has dimension 3.
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Actually in the smooth setting, the definition via derivations at x of smooth functions on the full manifold does work, although it may not be the most elegant way of doing things.
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The term "tangent map" appears in the Wiki list of missing math topics. Could somebody familiar with this article "Tangent space" please work it in? TIA --
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Question that doesn't belong on the talk page but I'm asking anyway because I don't give a rats ass in hell about any rule that limits my understanding:
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I found this term in an article about "how to be a graphics programmer". Should there be some mention of applications in this article?
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which provides an onward reference but I do not see much on
Knowledge about the subject - I guess I'm looking in the wrong place!
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You are absolutely right. Derivation takes a function and returns a value. I'm glad you pointed out the error. There was
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as though it is in the tangent space of the identity and then defining the exponential map from the tangent space of
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problematic edit I did not notice earlier because WP did not notify me (until now). So, that's a WP problem, too.
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is the same as using the exponential map at the origin and then "transporting" that transform over to
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Some formulas in the latter part of the article appear to be written in LaTEX, but they don't parse!
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The article says "all the tangent spaces of a manifold form another manifold of twice the dimension"
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on
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I didn't really read the whole article, but I noticed that it doesn't include the formula
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It is the vector space of derivations of the algebra of germs of smooth functions at x.
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The tangent space at x is not the vector space of derivations at x of smooth functions
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to "use" the tangent space at the origin everywhere. Apologies for the vagueness.
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Twice the dimension of WHAT? The original manifold? If so, please add that.
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In what sense does the tangent bundle have "twice the dimension" of something?
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Twice the dimension? Is that related to Jolt cola having twice the caffiene?
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for the sake of precision, since the definition should obviously involve
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This example was very helpful. Could it be inserted into the article?
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this ought to be included. I was just about to say the same thing.
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Please! We *can* do better than this atrocious opening paragraph.
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There is no mention of how this concept relates to the
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