If $z = f(x,y)$ has continuous partial derivatives then
$$dz = \frac{\partial z}{\partial x} \Delta x + \frac{\partial z}{\partial y}\Delta y$$
I have not seen the chain rule written this way before, but I am studying complex analysis and trying to prove the converse of Cauchy Riemann's equations is true if all the partial derivatives exist and are continuous. This form is used in the textbooks proof and I do not recognize it or understand where it comes from.
For $z = f(x(t), y(t))$, I'm used to seeing:
$$ \frac{dz}{dt} = \frac{\partial z}{\partial x}\frac{\partial x}{\partial t} + \frac{\partial z}{\partial y}\frac{\partial y}{\partial t} $$
What is the relationship between the two? And what does "$dz$" mean by itself?
dz is the total differential of z. I always try to think of it as measuring the total infinitesimal change in z due to both x and y. I like to read it as follows:
“dz= (the change of z wrt x)(a small change in x) + (the change of z wrt y)(a small change in y) “