Functions of Several Variables

A function of one variable $f$ is understood through the curve $y=f(x)$ in the $xy$-plane. A function $F$ of two variables $x,y$ is understood the same way, one dimension up: graph $z=F(x,y)$ in $xyz$-space. Given a point $(x,y)\in\mathbb R^2$, the value $z=F(x,y)$ is the height at which the corresponding point on the graph sits directly above (or below) $(x,y)$; the resulting set of points $(x,y,F(x,y))$ generally traces out a surface in three-dimensional space.

Worked example. Let $g(x,y)=30-x^2-y^2$ for $(x,y)\in\mathbb R^2$. Given the point $(2,3)\in\mathbb R^2$, the corresponding graph point is $\big(2,3,\,30-2^2-3^2\big)=(2,3,17)$, sitting $17$ units above $(2,3)$ in the $xy$-plane.

Slicing the surface — fixing one variable. If we fix $y=3$, the function $g(x,3)=30-x^2-9=21-x^2$ depends only on the single variable $x$; its graph, being quadratic, is a parabola — this is exactly the curve obtained by intersecting the surface $z=30-x^2-y^2$ with the plane $y=3$. Symmetrically, fixing $x=2$ gives $g(2,y)=26-y^2$, another parabola, the intersection with the plane $x=2$. The overall surface $z=g(x,y)=30-x^2-y^2$, every one of whose planar cross-sections (through a fixed $x$ or $y$) is a parabola, is called a paraboloid. …