Exploring Tensor Concepts with MIT OCW Courses

[3102]

Hello,

I am working through the MIT OCW courses 8.01 and 8.012. At my university we already learned about tensors in the first mechanics course but I don't really understand them completely.

Therefore I am searching for some MIT OCW course that covers tensors.
I'd be glad at any help.

Apart from that I've got some more questions:

1) When are tensors ordinarily covered in physics?

2) Could anyone give me some textbook recommendations for tensors?

 

[Khashishi]

When I was in school, tensors never really got covered as a dedicated subject. It just appeared from time to time, and we were more or less expected to just understand it. But it is typically covered in some detail in general relativity, which I never took. Most definitions of tensors revolve involve tensors in curved space, which is too complicated for someone who just wants to use tensors in mechanics. There are definitions which talk about how a tensor transforms under coordinate transformations. These definitions are mostly useless for your purposes. Since this is the majority of textbooks on tensors, I'm not going to recommend a textbook.

You know what a vector is right? A rank 2 tensor is simply a vector of vectors. A rank 3 tensor is a vector of vectors of vectors (or a vector of rank 2 tensors). Momentum is a vector. Now if you consider a fluid that carries momentum along with it, you can have a momentum flux. The fluid itself moves with some velocity, so you have a vector of vectors, or a rank 2 scalar. A vector in 3D has 3 components: vx, vy, vz. A rank 2 tensor in 3D also has three components, but each component is a vector, so there are 9 components total. It doesn't really matter if each fluid parcel carries only a little momentum and the fluid is moving fast, or each parcel carries a lot of momentum, and the fluid is slow; the momentum flux is the outer product of the fluid velocity and parcel momentum.

Often, you want to know the amount of a vector pointing in some direction. You would take the dot product of the vector and a surface normal (which looks like a vector but more properly should be called a covector), and you get a scalar (a rank 0 tensor). You can do the same for a rank 2 tensor, except that when you take the dot product of a rank 2 tensor with a covector you get a vector (a rank 1 tensor). You don't need to know all the stuff with contravariant and covariant vectors until you get into general relativity.