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I'm doing some surface integrals, and sometimes you have a surface given by let's say $z^2=x^2+y^2+1$. From the jacobian for $r = [x,y,f(x,y)]$ we have that it is equal to $[-\frac{\partial f}{\partial x}, -\frac{\partial f}{\partial y}, 1]$.

My book differentiates with respect to $x$ and $y$ and ends up with $\frac{x}{z}$ and $\frac{y}{z}$, respecively. Usually you're differentiating a function say $f(x,y) = something$ and you're only looking at "something" (obviously). What is the logic when you have $z = something$. Wouldn't the derivative of z with respect to x just be 0 anyway?

My attempt would yield $0 = 2x$ which is obviously incorrect. Can someone help me understand the right way to interpret this?

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