What Does Equation 5.7 in Introducing Einstein's Relativity Really Mean?

[Corbeau]

Hello. I'm going through Ray D'Inverno's "Introducing Einstein's Relativity" and I'm stuck at a certain point and can't move forward. It deals with tensors, I'm stuck at the transformation matrix and the problem is, I can't figure out what the key equation (5.7) actually means. There is a screenshot of the part of the page below, and I have no idea even how to start understanding it. What is it? What was it derived from? How do you actually use it? The left-hand side looks like a 1x1 matrix and the right-hand side looks like a n x n matrix and, of course, that wouldn't make any sense. If someone could write a few helpful sentences or point me in the direction of an Idiot's Guide to Tensor Differentiation, I would be most grateful. Thank you.

Screenshot:

 

[DEvens]

It's just some lazy notation. Note that the top x has an "a" superscript, and the bottom x has a "b". So they run over 1 to n. So it's just saying that the a index runs "down" in the matrix on the right side. And the b index runs "right."

 

[George Jones]

I can't figure out what the key equation (5.7) actually means. There is a screenshot of the part of the page below, and I have no idea even how to start understanding it. What is it? What was it derived from? How do you actually use it? The left-hand side looks like a 1x1 matrix and the right-hand side looks like a n x n matrix and, of course, that wouldn't make any sense.

The left side is also nxn, as both indices a and b range independently from 1 to n.

Have you studied multivariable calculus? If so, wasn't the Jacobian covered?

 

[WWGD]

Express one variable as a function of the other, i.e.,$f^:=(g_1(x_1,..,x_n), g_2(x_1,..,x_n),...,g_n(x_1,x_2,..,x_n))$ then the matrix representing the Jacobian, i.e., the matrix M where $(m_{ij}):= \frac {\partial g_i}{\partial x_j}$ is the matrix in 5.7. You may want to work out a specific example like , e.g., changing from Cartesians to Polars ( of course, this can only be done locally) and backwards.

 

[George Jones]

You may want to work out a specific example like , e.g., changing from Cartesians to Polars ( of course, this can only be done locally) and backwards.

Exercise 5.2 in the book from which Corbeau is studying asks for the Jacobian for Cartesian to spherical.

 

[WWGD]

Wow, George, you seem to own a lot of books. Or, by George, you seem to own a lot of books.

 

[Chestermiller]

This is not a tensor differentiation. This describes how you transform the components of a tensor from one coordinate system to another coordinate system.

A vector is a first order tensor. Suppose you know the components of the vector in Cartesian coordinates, and you want to find the components of the same vector is cylindrical coordinates. In other words, the original components are expressed with respect to the x, y, and z unit vectors, and you want to find the components expressed with respect the the r, θ, and z unit vectors. That's an example of what this is all about.

Chet

 

[Corbeau]

Warm thanks to everybody, I managed to push over this particular bump thanks to some advice here. The problem is I'm encountering stuff like that on every other page, but that's a different issue; I've finished my student era a while ago and now need additional time and effort to recall all the tools and tricks that are needed here. Well, thank gods for the internet and infinite sources :)

 

[Fredrik]

You should check out "Linear algebra done wrong" by Sergei Treil. It's free, it's the best book on linear algebra (based on the 60% of it that I've read so far, as well as other people's comments), and it contains a chapter on tensors (that I haven't read yet).

Another very good way to get introduced to tensors (and special relativity) is to read the first three chapters of "A first course in general relativity" by Schutz.