Don't understand what the book means, a tensor thing....

[Oz123]

Homework Statement

Right, so it's not really an assignment or anything, just confused of what a book says. the book is "mathematical methods for physicists." The screenshot is attached.
The thing that I'm confused about is that it says "As before, aij is the cosine of the angle between x′i and xj ." This is from rotating the coordinates, so x' is the new coordinates whilst x is the old one. And then it defined aij as the partial derivatives of the x'i wrt xj...Now I don't know how the cosine of the angle between x'i and xj is equal to the partial derivative. Can anyone explain this to me? Thank you in advanced!

 

[Oz123]

Here's more info about aij being cosine angles:

 

[pasmith]

If $$x'_i = \sum_j a_{ij} x_j$$ then $$\frac{\partial x'_i}{\partial x_k} = \sum_j a_{ij} \frac{\partial x_j}{\partial x_k} = a_{ik}$$ since $$\frac{\partial x_j}{\partial x_k} = \delta_{jk}$$.

 

[Oz123]

Thank you very much!