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To understand partial derivatives geometrically, we need to interpret the algebraic idea of fixing all but one variable geometrically. This is equivalent to slicing a surface by a plane to produce a curve in space.
All the same ideas carry over in exactly the same way to functions $f(x,\,y,\,z)$ of three or more variables  just don't expect lots of pictures!! The partial derivative $f_z$, for instance, is simply the derivative of $f(x,\,y,\,z)$ with respect to $z$, keeping the variables $x$ AND $y$ fixed now. Information about the partial derivatives of a function $f(x,\,y)$ can be detected also from the contour map of $f$. Indeed, as one knows from using contour maps to learn whether a path on a mountain is going up or down, or how steep it is, so the sign of the partial derivatives of $f(x,\,y)$ and relative size can be read off from the contour map of $f$.
