Vtyjkyuk

What is the chain rule of a multi-variable function with arguments that are also multi-variable functions?

Suppose $x$, $y$, $z$ are independent variables. I mean changing $x$ won't change $y$ and $z$.

Is the general form of multi-variable chain rule similar to the following?

$$fracpartial w(u(x, y), v(y, z), t(y, z))partial x = fracpartial wpartial ucdotfracpartial upartial x + fracpartial wpartial vcdotfracpartial vpartial x $$

Thanks.

That's nearly right, but you left off the third term that accounts for $t$. So you should have
$$fracpartial wpartial x = fracpartial wpartial ucdotfracpartial upartial x + fracpartial wpartial vcdotfracpartial vpartial x + fracpartial wpartial tcdotfracpartial tpartial x$$

where $w$ is a function of $u,v,t$ and $u,v,t$ are functions of $x,y,z$.

One mnemonic device for remembering this is to think of it as summing over all products of "fractions" that will partially cancel and each give $partial w$ in the numerator and $partial x$ in the denominator. But be warned that that's not really what's happening, these aren't fractions and they aren't cancelling. Nonetheless, this helps you to remember it.

Addendum: Of course, if $x$ doesn't actually appear in the formula for $u$, $v$, or $t$, then that partial is zero. What I wrote is the most general formula, not just the formula for your specific case. In other words, even if $v=v(y,z)$ , you can still think of it as $v(x,y,z)$; but then $partial v/partial x = 0$ , and similarly for the others. Not sure if you really wrote what you meant, but my answer is applicable to the general case.

beginalign
fracpartial w(u(x, y), v(y, z), t(y, z))partial x &= fracpartial wpartial ucdotfracpartial upartial x + fracpartial wpartial vcdotoverbracefracpartial vpartial x^0+ fracpartial wpartial tcdotoverbracefracpartial tpartial x^0 \
&=fracpartial wpartial ucdotfracpartial upartial x,
endalign
or what I think you meant to write
beginalign
fracpartial w(u(x, y), v(colorredx, z), t(y, z))partial x &= fracpartial wpartial ucdotfracpartial upartial x + fracpartial wpartial vcdotfracpartial vpartial x+ fracpartial wpartial tcdotoverbracefracpartial tpartial x^0 \
&=fracpartial wpartial ucdotfracpartial upartial x+ fracpartial wpartial vcdotfracpartial vpartial x.
endalign