What is Tensor Calculus?

1. Jun 2, 2004

QuantumDefect

Im only in second semester calculus and my friend keeps on babbling about Tensor Calculus and how only a few people know how to do it in the world. I highly doubt that only a few people in the world know how to do this because there are plenty of math graduates out there, as well as professors. Can anyone tell me what exactly it is and the problems its meant to solve? Thank you.

2. Jun 2, 2004

HallsofIvy

If you look closely you will see a sub-forum titled "Tensor Analysis and Differential Geometry" just three places down from this. A "tensor" is a generalization of vector. It can be applied to just about any kind of problem that vectors can. As long as you are dealing with Euclidean space (curvature 0) you might as well just use vectors but problems involving curved surfaces or spaces (such as general relativity) use tensors.

3. Jun 2, 2004

e(ho0n3

As HallsofIvy said, you should look in the Tensor Analysis and Differential Geometry forum. If you take Calculus III or Advaced Calculus you'll find out what they are. As HallsofIvy said as well, tensors are an 'abstraction' of the whole concept of vectors. I've used tensors to solve some problems in mechanics, like when dealing with stresses on some object (where you have to work with a so-called stress tensor).

e(ho0n3

4. Jun 2, 2004

quartodeciman

What is often called Tensor Calculus was called Absolute Differential Calculus back at the start of the twentieth century and then existed only in mathematics research publications. Only a few physicists and mathematicians knew much about it. Albert Einstein was introduced to it by his school-chum Marcel Grossmann, when AE was stuck trying to develop a relativistic theory of gravitation that generalized transformations beyond the Lorenz Transformation. It utterly altered the way AE did fundamental scientific research, making him more bound than before to matters of mathematical form.

The Absolute Differential Calculus was mainly developed by Gregorio Ricci-Curbastro and Tullio Levi-Civita.

http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Ricci-Curbastro.html

http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Levi-Civita.html

http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Grossmann.html

After AE's success with General Relativity and his subsequent fame, many physicists and mathematicians mastered this subject and applied it to other problems. Therefore it is included in the curriculum for Mathematical Physics.

5. Jul 23, 2004

mathwonk

I don't really know much about tensors but this is anonymous so I can say what I think without fear. In algebra a tensor is just a way of multiplying vectors. So the dot product is one example of a tensor. hence you have already seen a tensor.

Moreover note that a taylor series approximates a function by polynomials of various orders, of which the first order term is linear, i.e. a vector. The second and higher order terms are bilinear, and trilinear, and so on, since they involve multiplying 2 or 3 or more vectors.

Recall that the curvature of a curve involves the second derivative. Thus in higher dimensions, curvature is a tensor. E.g. the curvature of a surface is a sort of product defined on tangent vectors to the surface.

I hope this is not too far wrong.

6. Jul 23, 2004

chroot

Staff Emeritus
7. Jul 23, 2004

mathwonk

Thank you chroot! I am certainly enjoying myself.

8. Oct 16, 2009