Chain Rule in Multiple Variables

Question

Say I have a function $$z(s,t,x,y)=f(s(x(t,s),y(t,s)),t(x(t,s),y(t,s)))$$

How would I find $$\frac{\partial z}{\partial x}$$

The chain rule here is confusing me.

Answer 1

The notation $\frac{\partial}{\partial x}$ can be ambiguous. Or it means $$\frac{\partial z}{\partial x}(s,t,x,y):=\lim_{h\to 0}\frac{z(s,t,x+h,y)-z(s,t,x,y)}{h}$$ provided that the limit exists, that is it is the derivative in the $3$-rd variable (and I would prefer the notation $\partial_3$); or it means $$\frac{\partial z}{\partial x}(s,t,x,y):=\lim_{h\to0}\frac{z(s(x+h),t(x+h),x+h,y(x+h))-z(s(x),t(x),x,y(x))}{h}$$ provided that the limit exists, that is you compute the total derivative of $z$ in the direction $x$ when $s,t$ and $y$ depends on $x$.

In general, $\frac{\partial}{\partial x}$ denotes the partial derivative in the $x$ coordinate (here in the $3$-rd variable), the last situation above being written as $\frac{\mathrm{d} z}{\mathrm{d} x}$ i.e. the coefficient in front the coordinate linear form $\mathrm{d} x$ in the expression of the differential $\mathrm{d} z$.