What is Tensor Calculus and Its Applications in Mechanics?

[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.

 

[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.

 

[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

 

[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.

link to R-C bio --->
http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Ricci-Curbastro.html

link to L-C bio --->
http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Levi-Civita.html

link to Gr bio --->
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.

 

[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.

 

[chroot]

Read this thread mathwonk:

https://www.physicsforums.com/showthread.php?t=35920

Welcome to PF, by the way.

- Warren

 

[mathwonk]

Thank you chroot! I am certainly enjoying myself.

 

[MadRocketSci2]

Ah, the old "I know something you don't know" game.

Tensors are pretty ubiquitous in continuum mechanics. I haven't yet pursued tensor calculus into the highly formalized and detailed world that mathematicians use, or that you have to know to hack differential geometry, but I've worked with stress tensors, strain tensors, moment's of inertia (which can be summarized as a tensor for 2+ dimensional objects).

A (rank 2) tensor is a generalized mathematical object that linearly relates a vector to another vector. A rotation matrix is a tensor. It returns the rotated vector for an input vector. A stress tensor would return a force vector for a differential area normal vector inside an object.

Where the complexity starts coming into play is when you start messing around with transforming your basis with more complicated tensor expressions.