# Divergence in orthogonal curvilinear coordinate system question

I'm trying to understand how the divergence formula in curvilinear coordinates is derived, but unfortunately my textbook doesn't go into much detail. Here is what they show: I think I understand how they get the LHS of (8.9) but I don't know how they get to the RHS. Can someone please show me how the LHS equals the RHS?

## 1 Answer

The expression you are considering is the definition of differential for a function obscured by the notation the author is (forced) to use. You can see it in one variable:

$$f(x+dx)=f(x)+f'(x)dx +[\text{terms of second and greater degree}]$$

so is, $f(x+dx)\approx f(x)+f'(x)dx$ or $f(x+dx)-f(x)\approx f'(x)dx$

Now, simply "translate" it considering that partial derivatives are the same as as derivatives for a single variable maintaining constant the variables not involved in the derivative:

$A_1h_2h_3|_{x^1x^2x^3}$ is some function of the three variables, say $g(x^1,x^2,x^3)$, then, we, following more strictly than the author the notation he chose, can write: $\left.\dfrac{\partial(A_1h_2h_3)}{\partial x^1}\right|_{(x^1,x^2,x^3)}=\dfrac{\partial g(x^1,x^2,x^3)}{\partial x^1}$, so is, the derivative wrt $x^1$ evaluated at the point $(x^1,x^2,x^3)$

You surely can complete the "translation" (remember that $x^2$ and $x^3$ act as simple constants here).