Poisson equation

Question

For the equation $\frac{f(x+he_i-y)-f(x-y)}{h}$, I am wondering whether we should treat $y$ is constant here?

Answer 1

My interpretation is that you are unsure what $$\frac{f(x+he_i-y)-f(x-y)}{h}\to f_{x_i}(x-y)$$ is supposed to mean (please comment if this is wrong!)

In this expression, we are treating both $x$ and $y$ as constant, while sending the $h$ to $0$. More formally the line could be written out as

$$\lim\limits_{h\to 0}\frac{f(x+he_i-y)-f(x-y)}{h}= \frac{\partial f}{\partial x_i}(x-y),$$ i.e. the expression goes to the directional derivative of $f$ evaluated at the point $(x-y)$. This is true because the left hand side is in fact the limit definition of the derivative.