What Are Tensors and Their Applications in Mathematics?

[nicksauce]

I often hear people talking about tensors, and furthermore I think I have used them without knowing it in some classes (Levi-Civita symbol for example), but I have never been able to get a good explanation of what a tensor is or when they are used.

 

[cristo]

This question has been answered loads of times: try here or here. You could also try having a read of some external websites, like this (http://mathworld.wolfram.com/Tensor.html). Feel free to ask anything more once you've read a little about the basics.

 

[HallsofIvy]

A "real" explanation would hardly fit here but a very rough idea is: a tensor is the most general thing that changes "homogeneously" as we change coordinate systems: That is, if A is a tensor with components written in terms of one coordinate system and A' is the same tensor with components written in terms of another coordinate system, then A'= PA where P depends only on the two coordinate systems. The crucial point of that multiplication really is the P(0)= 0: if a tensor, A, is represented by "all 0 components" in one coordinate system, then it is represented by all 0 components in any coordinates system. If we have some "physical law" represented by the equation A= B, where A and B are tensors, the saying it is true in one coordinate system says that A- B= 0 in that coordinate system and so A- B= 0, which gives A= B, is true in any coordinate systems. Since physical laws should be independent of the arbitrarily chosen coordinate system, tensors are the "natural" way to express them.

 

[Andy Resnick]

I have difficulty with abstract definitions of mathematical objects. Specifically for the stress tensor, I picture the surface of a cube. Each surface has three orthogonal directions associated with it- one perpendicular to the surface and two that lie in-plane. The normal components correspond to pressure/dilation/compression and are the diagonal elements of the tensor. The in-plane components correspond to shear, and act to deform a square into a parallelogram. These are the off-diagonal terms.

 

[symbolipoint]

That is some wild stuff. Would engineers be far more likely to use tensors than would any other mathematically proficient professional person? I also notice that some undergraduate mathematics programs seem to have options which apparently do not stress a course on tensors (although I may be mistaken, since although courses in some options do not contain reference to tensors in their descriptions, maybe some actual reference to tensors are done during course itself).

 

[HallsofIvy]

Engineers certainly do use tensors, though probably "Euclidean" tensors where you assume all coordinate systems have straight line axes, at right angles to each other (which does NOT include polar coordinates, spherical coordinates, or cylindrical coordinates). Physicists, especially those working in General Relativity would use more general ternsors.

 

[genneth]

Do engineers ever work with tensors in more than 3 dimensions or over rank 2 and symmetric in some/all of the indices? Because I'd love it if someone could give a decent visualisation of those...

 

[mathwonk]

you know what a vector space is. there are two aspects to tensors, one involves multilinear combinations of vectors, and multilinear functions on a vector space, and the other involves forming families of these objects. let us call the first type tensors, and the second type tensor fields.

thus a family of tangent vectors to a surface is a tensor field. but also a family of multilinear functions on tangent vectors is a tensor field, e.g. a family of dot products on tangent vectors is a tensor field.a tensor is a sum of things like v.w...u.f.g...h, where each v,w,u is a vector, and each f,g,h is a linear function on vectors. a tensor field is a family of such things.