#math/calculus A one-variable function assigns an output to each point on a line. Allowing two independent inputs turns the line into a plane and, for a scalar output, turns the graph into a surface. A function of two variables is a map $ f:D\subseteq\mathbb R^2\to\mathbb R, \qquad (x,y)\mapsto f(x,y). $ Its graph is the surface $ \{(x,y,z):z=f(x,y),\ (x,y)\in D\}\subseteq\mathbb R^3. $ The domain is part of the function. For $ f(x,y)=\frac{x}{x^2+y^2}, $ the natural domain is $\mathbb R^2\setminus\{(0,0)\}$. A formula alone can hide such exclusions. A function of three variables, $F(x,y,z)$, assigns a number to points in space. Its graph would require four dimensions, so we usually visualize it by [[14C2 - Level Curves and Level Surfaces|level surfaces]] instead. Think of $f(x,y)$ as temperature over a map, height over terrain, or loss as two parameters vary. This turns a static surface into the object whose local change we study in [[15A1 - Partial Derivatives|partial differentiation]].