What does the $\partial$ notation means, in tensor analysis?

Question

I have seen in a lot of introductory texts in Tensor analysis (Arfken & Weber's Mathematical Methods, for example). The last one i saw here (Unfortunately, this text is only avaliable in portuguese) (the author uses the notation in page 216) is that if you have a matrix equation between two vectors (in this example is a rotation matrix): $$x'_i = L_{ij}x_j$$ Then, you can write $$L_{ij} = \frac{\partial x'_i}{\partial x_j}$$ This, for me, doesn't seems like a partial derivative. If it is, then why and how? If it is not, then what it is?

Answer 1

It is indeed a partial derivative.

For example, you could ask the question "if I use a different set of three coordinates $x',y',z'$ to describe a point in space which in the old coordinates was $(x,y,z)$, and then I change $x$ to some $x+\delta x$, how do each of $x',y',z'$ change? Notice that in this question, I change $x$ keeping $y$ and $z$ fixed; thus the partial derivative.

(What I said describes in fact a covariant tensor; the relation you give describes a contravariant tensor, but the idea is the same.)

In the case of a specific rotation, these nine partial derivatives form a (constant) matrix $L_{ij}$ but for different types of transformations, the transformation could easily depend on where you are in space.