8 ms·

It has nothing to do with tensor fields, uniform/constant tensors still obey the proper coordinate transformations, that's the defining property of any tensor.

• catgary 5y ago
  What do you think the tensor product of finite dimensional vector spaces looks like?
• contravariant 5y ago
  Mathematicians would disagree with you there. There are no coordinates to transform in an ordinary tensor space and therefore no way for a tensor to be affected by such a transformation. Matrices (or linear transformations in general) are important examples of tensors. There's a nice adjunction between tensor spaces A(x)B and the space of linear transformations B=>C given by: Hom(A(x)B, C) = Hom(A, B=>C) In the case of Tensorflow I think they do actually still talk about linear transformations of some kind so it's perfectly fine to call them tensors.
  • cygx 5y ago
    There are no coordinates to transform in an ordinary tensor space and therefore no way for a tensor to be affected by such a transformation. Sure there are: Any basis of the underlying vector space(s) induces a basis of the tensor space. Components respective to some basis are coordinates. You can then investigate what happens to the induced basis (or rather, the respective components) under a basis transformation of the underlying vector space(s), which is where the "physicist's" definition of tensors originates.
    • contravariant 5y ago
      The components of a vector aren't the same as the coordinates physicists talk about when they're dealing with tensors. The components would be something like the value of the magnetic potential, or the local wind speed. The coordinates would be the location where that particular vector is 'anchored'. A change of coordinates does indeed induce a change of basis, but a change of basis isn't really a change of coordinates. And strictly speaking some vector spaces don't really have an obvious basis (without invoking choice), so having a basis be a prerequisite for the definition is not ideal. The whole requirement that a tensor is 'something that transforms like [...] under a coordinate transformation' is just how physicists have chosen to phrase that a vector bundle is only well defined if it's definition isn't dependent on some arbitrary choice of coordinates. In my opinion this requirement is more easily apparent in the mathematical definition where there is no choice of coordinates in the first place, rather than the physicists way of working with some choice of coordinates and checking how things transform.
      • cygx 5y ago
        I'm aware. Though if we want to be more precise, that's about tensor fields, where the basis transformations of the underlying vector bundles (the tangent and cotangent bundle) are in turn induced by coordinate transformations of the base manifold. However, physicists get introduced to tensors far earlier than any excursions into differential geometry when discussing rigid bodies.
        • contravariant 5y ago
          Yes I'd also call those tensor fields. The main point I'm trying to make is that the tensor transformation law only makes sense for such fields.
          • cygx 5y ago
            The terms co- and contravariant make sense on a purely algebraic basis, with components of tensors transforming 'the same as' or 'opposite to' the basis vectors. That the basis transformation is induced by transformations of some base manifold is incidental.
  • tagrun 5y ago
    Which mathematician disagrees with what exactly? Tensors are introduced by physicists to ensure various physical quantities (which involve coordinates and their derivatives) do not depend on the arbitrarily chosen coordinate system. This is ensured through the transformation properties of tensors. The name tensor itself comes from the theory of elasticity, Cauchy stress tensor, which BTW is uniform in many practical cases, and obeys the following tensor transformation rule: https://en.wikipedia.org/wiki/Cauchy_stress_tensor#Transformation_rule_of_the_stress_tensor https://en.wikipedia.org/wiki/Cauchy_stress_tensor#Transform... like any other (contravariant) tensor must. Matrices are not examples of tensors. Matrices can be used for representation of tensors, in which case tensor product becomes Kronecker product, but matrices in general don't have to represent tensors. You can put anything, including your favorite colors or a list of random numbers, in a matrix, and it won't be a tensor in general, not unless it must transform like a tensor under coordinate system changes. Similarly, TensorFlow "tensor" is just a multidimensional data array, with no transformation rules enforced on it, and therefore is not a tensor.
    • vcxy 5y ago
      Interesting, you use the word matrix differently than me. The way I use it, a 2d array isn't necessarily a matrix. It's only a matrix if it represents a linear map between vector spaces with respect to chosen bases. Then again, I'm not much of a programmer, but I've taught linear algebra a few times. My head is just in a different place I guess.
      • tagrun 5y ago
        Any 2D array of data is a valid matrix. But not any multidimensional data is a valid tensor.
        • vcxy 5y ago
          I'm not sure I buy this after you just used an example of a 2d array of you're favorite colors just a few minutes ago. Maybe I'm missing something. What kind of linear transformation does that represent, and between what vector spaces?