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639:(with multiple polytopes being assigned to each point). Further methods are available, but every visualization method must make compromises, and in general higher-order tensors are notoriously difficult to visualize. I can understand why this would be the case much easier if the geometric complexity of the units of representation grows as the order grows. But if a 3rd-order tensor can be parametrized by three vectors, what then is the point of using the higher-order form? ᛭
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495:-dimensional manifold. The question is, what is the geometric meaning of such a thing? It is not obvious that geometrically taking square roots is harmless (it isn't for complex numbers). But you are right that it is time to support this all by references (Guillemin-Sternberg, Geometric Asymptotics, p. 251).
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A tensor field is almost universally (or at least in physics) assumed to be a function from the manifold to the tensor algebra of the tangent space. It is possible to have a tensor bundle on a manifold that is not related to the tangent space. The description, as given in this article, makes little
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By analogy with the tensor field arrows, shouldn't the vector field illustration have 2 arrows for each cell. One arrow pointing out of the cell and one in the cell and their product is the resulting arrow? I think that is the sense of the final illustration. Each pair of arrows defines a product.
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My understanding (and this is more from intuition) is that in a plot of tensors any number of arrows greater than one represent a 2nd-order tensor of that dimension. Note that there are three arrows with three degrees of freedom; the tensors in question can be represented by 3x3 matrices, with each
706:
I am trying to teach myself the basics of fluid mechanics in order to implement wind flow in a video game, and I have been intimidated all night by the constant references to tensors. The sentence in the introduction about how a tensor is a generalization of a scalar field and a vector field was
669:). Anyway, the graphic was supposed to illustrate a 0th order, 1st order, and 2nd order tensor in two dimensions. But 2nd order tensors in two dimensions are represented in a basis by 2×2 matrices, not 3×3 matrices, and so it would be more natural to display a
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of vectors at each point. In any event, if there is to be such a graphic, then it would be better to also explain the meaning of the representation of a 2nd order tensor as a pair of vectors, lest readers leave with the false impression that an
904:. Avoid vague sentence like that. Almost no readers will have any idea what "a vector space ... depending on an angle" means. And nobody will understand how it might be possible that "a vector space ... could look like a Möbius strip."
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vector representing a separate three indices. To represent a 3rd-order tensor requires groups of such arrows (multiple 2nd-order tensors). To differentiate you could use color coding or make them the vertices of a
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could just as easily be a vector space. It could be that I'm overlooking some subtleties, but if so, I'd love to hear what they are. (Perhaps it would be good in general to add one or two references.) --
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This article was automatically assessed because at least one WikiProject had rated the article as start, and the rating on other projects was brought up to start class.
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exactly what I needed. I'm not scared anymore. That is a much simpler and more concise explanation than is found in the actual article on tensors.
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This article is part of a series of closely related articles for which I would like to clarify the interrelations. Please contribute your ideas at
661:, and as you can see I have removed the graphic. For one thing, 3rd-order tensors are not parametrized by three vectors (you are confusing the
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I finally figured out what this means, but for anyone who is unfamiliar with fibre bundles, this will make
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The section on "twisting by a line bundle" and "half-density" is quite opaque as it currently is written.
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which redirects here again, and there is no actual definition of a tensor bundle given in the section. --
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reference to the tangent space, and is thus more general. Is the more general definition intended? —
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I'm planning on fixing this, but at the rate I'm going, it'll take another 6 months. Mean time
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https://math.stackexchange.com/questions/270297/difference-between-tensor-and-tensor-field
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The difference is all in your head. Literally. The difference in calling the same object
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is a module over a ring, instead of a vector space over a field? It seems to me that
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The opening to this article is therefore incorrect. The
Riemann curvature tensor
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850:. But this is not even continuous on the real numbers, i.e.
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OK, a half-form would be something like a square root of an
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Is there any particular reason that it only assumes that
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