Functions of Several Variables, Limits, and Continuity

Single-variable calculus studies quantities that depend on one input. Most quantities depend on several. The temperature on a metal plate depends on both coordinates of a point; the volume of a cylinder depends on radius and height; the wind-chill index depends on temperature and wind speed. Differentiating and integrating such quantities rests on two objects: the functions themselves and the limit that underlies calculus in more than one variable.

Functions of two variables

When a function is given by a formula with no stated domain, the domain is the set of all pairs for which the formula returns a real number.¹

A function of three variables assigns a number $f(x,y,z)$ to each point of a region $D\subseteq {\mathbb{R}}^3$, and a function of $n$ variables assigns $f(x_1,\dots ,x_n)$ to each point of $D\subseteq {\mathbb{R}}^n$.

Graphs

A single-variable graph is a curve in the plane; a two-variable graph is a surface in space. The graph of a linear function $f(x,y)=ax+by+c$ is a plane. The graph of $f(x,y)=x^2+y^2$ is a bowl (a circular paraboloid); the graph of $f(x,y)=\sqrt{9-x^2-y^2}$ is the upper half of a sphere of radius $3$.

Drawing a surface by hand is hard, and reading three-dimensional structure off a flat page is harder. Two flat devices carry most of the information: traces and level curves.

A trace is the curve where the surface meets a plane. The vertical trace in the plane $x=a$ is the curve $z=f(a,y)$; the vertical trace in $y=b$ is $z=f(x,b)$. Horizontal traces $z=k$ are the level curves.

Level curves and contour maps

A contour map draws several level curves $f(x,y)=k$ for equally spaced values of $k$. Reading it is reading a topographic map: each curve joins points of equal height, and the surface is steep where the curves are close together, gentle where they are far apart.

The surface and its contour map encode the same information.

Functions of three or more variables

A function of three variables cannot be graphed (its graph would live in ${\mathbb{R}}^4$), so it is studied through its level surfaces.

For $f(x,y,z)=x^2+y^2+z^2$ the level surfaces $x^2+y^2+z^2=k$ are concentric spheres of radius $\sqrt{k}$ (for $k>0$), a nested family that fills space. A level surface is the three-variable analog of a level curve, one dimension up.

Limits of two-variable functions

The single-variable limit ${\lim }_{x\to a}f(x)=L$ requires the left limit and the right limit to agree; $x$ can approach $a$ from only two sides. In the plane, $(x,y)$ can approach $(a,b)$ along infinitely many paths, and the limit must give the same value along all of them.

The quantity $|f(x,y)-L|$ is the distance between output and target; $\sqrt{(x-a{)}^2+(y-b{)}^2}$ is the distance from $(x,y)$ to $(a,b)$. So the definition says the output can be forced within $\epsilon$ of $L$ by keeping the input within some disk of radius $\delta$ about $(a,b)$ — but not at the center itself.

The multi-directional nature of the plane gives a decisive negative test.

The two paths need not be the coordinate axes.

A line-independent value along every line is still not enough.

To prove a limit exists, bound $|f(x,y)-L|$ by a quantity depending only on the distance $r=\sqrt{x^2+y^2}$ that vanishes as $r\to 0$, then apply the Squeeze Theorem.

Continuity

Continuity means small changes in the input produce small changes in the output: the graph has no hole, tear, or jump. Because limits of sums, differences, products, and quotients behave the way they do in one variable, the standard constructions stay continuous.

• Polynomials. A polynomial in $x$ and $y$ is a sum of terms $c{x}^m{y}^n$ ($m,n$ nonnegative integers). Every polynomial is continuous on all of ${\mathbb{R}}^2$.
• Rational functions. A ratio of polynomials is continuous everywhere its denominator is nonzero.
• Compositions. If $g$ is continuous at $(a,b)$ and $h$ is continuous at $g(a,b)$, then $h\circ g$ is continuous at $(a,b)$. Feeding a continuous two-variable function into $\sin$, $\exp$, or a root keeps it continuous.

Continuity in three or more variables reads the same way, ${\lim }_{(x,y,z)\to (a,b,c)}f(x,y,z)=f(a,b,c)$, with distance measured in ${\mathbb{R}}^3$. All of it collapses to the one-variable case when the extra inputs are held fixed — the idea behind the partial derivative, which slices a surface with a plane and differentiates the resulting curve.