Books about tensor analysis just good enough for physics

[Haorong Wu]

Hi. I am looking for a book about tensor analysis. I am aware that there have been some post about those books, but I wish to find a thin book rather than a tome but just good enough for physics, such as group theory, relativistic quantum mechanics, and quantum field theory.

I am reading Mathematical Methods for Physicists by Arfken, Weber, Harris. The content in the chapter of tensors is quite easy to understand, but when it comes to the exercises, I can only solve half of them. Besides, even looking at the solutions, I can not understand them.

For example,

If $T_{...i}$ is a tensor of rank $n$, show that $\partial {T_{...i}} / \partial {x^j}$ is a tensor of rank $n+1$. (Cartesian coordinates).

The solution is just a sentence that

As the gradient transforms like a vector, it is clear that the gradient of a tensor field of rank $n$ is a tensor of rank $n+1$.

I cannot understand the solution at all. Maybe the content of this chapter is not self-content?

By the way, what would you do if the problems in some book are hard to solve? I have read nonlinear optics by Boyd and quantum optics by Scully. I find that I can handle the content of these two books, but I can not solve most of the problems. My tutor ask me to just read the books, and leave the problems aside. However I do not feel good enough if I cannot solve problems. Maybe I have some problems...

 

[anuttarasammyak]

Not as a correct mathematics but to be accustomed to the way, I will show
$$\frac{\partial T_{abc...hi}}{\partial x^j }=T_{abc...hi \ \mathbf{,\ j}}$$
It shows that differentiation by contra variant component is one more addition to covariant component and "," means this differentiation or gradient.
For an example of rank 0 tensor, scalar $\phi$
$$\frac{\partial \phi}{\partial x^i}=\phi_{,i}=-\mathbf{E}$$
minus electric field vector (= rank 1 tensor of covariant component ) in electrostatics for i={1,2,3}.

 

[weirdoguy]

vector (= rank 1 tensor of covariant component )

Standard nomenclature is that it's covector, not vector. But you can transform it to a vector using metric tensor.

 

[Haorong Wu]

Not as a correct mathematics but to be accustomed to the way, I will show
$$\frac{\partial T_{abc...hi}}{\partial x^j }=T_{abc...hi \ \mathbf{,\ j}}$$
It shows that differentiation by contra variant component is one more addition to covariant component and "," means this differentiation or gradient.
For an example of rank 0 tensor, scalar $\phi$
$$\frac{\partial \phi}{\partial x^i}=\phi_{,i}=-\mathbf{E}$$
minus electric field vector (= rank 1 tensor of covariant component ) in electrostatics for i={1,2,3}.

Thanks! But I still do not understand the solution.

By the way, I have finished tensor analysis, pseudotensors, dual tensors, tensors in general coordinates. the remaining part of the chapter have Jacobians, differential Forms, differentiating forms, and integrating forms. Should I finish the remaining part? I do not see the connections between them and those subjects that I am going to learn, such as group theory, quantum field theory, and relativistic quantum mechanics.

If they are not relavent to those areas, I would direct my limited time to other chapters.

 

[anuttarasammyak]

Should I finish the remaining part?

I am not qualified to the fields you mentioned, but I recommend you to go through the text anyway. You will be more confident on the mathematics and get a place to come back when you really need it.

 

[George Jones]

I am reading Mathematical Methods for Physicists by Arfken, Weber, Harris. The content in the chapter of tensors is quite easy to understand, but when it comes to the exercises, I can only solve half of them. Besides, even looking at the solutions, I can not understand them.

For example,The solution is just a sentence thatI cannot understand the solution at all. Maybe the content of this chapter is not self-content?

My copy of Arfken and Weber has decided to self-isolate at a location different from my location, so I do not have it at hand. Probably Arfken, Weber, and Harris define tensors as quantities that have certain transformation properties under a change of coordinates. (It is difficult for me to imagine that Arfken, Weber, and Harris define tensors as multilinear maps.)

As an example, consider a tensor (with components) $T_{ab}$. Define a new quantity $A_{abc}$ by
$$A_{abc} = \frac{\partial T_{ab}}{\partial x^c}.$$

By considering a new coordinate system, and by using the multivariable chain rule, show that $A_{abc}$ has the right transformation property to be a tensor.