Functions of Several Variables, multivariable and vector calculus lesson

Hold a thermometer at some spot on a heated metal plate and it shows one number. Move it and the number changes. The reading depends on two inputs, the coordinates of the spot, and gives one output. Many quantities in science work like this, and they need a language of their own before any calculus is done with them.

Key terms

Domain and range

Definition A function of two variables is a rule $f$ that assigns to each point $(x, y)$ of a set $D \subseteq \mathbb{R}^2$ exactly one real number $f(x, y)$ . The set $D$ is the domain of $f$ and the set of all values

\{\, f(x, y) \mid (x, y) \in D \,\}

is its range. A function of three variables is defined the same way with $D \subseteq \mathbb{R}^3$ . When only a formula is given, the domain is understood to be the largest set of points where the formula gives a real number, so square roots need nonnegative arguments, logarithms need positive arguments and denominators must not be zero.

In plain words The domain is every input point where the formula makes sense. The range is every number that actually comes out.

Example Take

f(x, y) = \sqrt{4 - x^2 - y^2}.

The square root needs $4 - x^2 - y^2 \ge 0$ , so the domain is the closed disk $x^2 + y^2 \le 4$ of radius $2$ . On that disk the quantity under the root runs from $0$ (on the circle) up to $4$ (at the origin), so the range is $[0, 2]$ . Every value $c$ in $[0, 2]$ really occurs, for example at the point $\left(\sqrt{4 - c^2}, 0\right)$ .

Scalar field

Definition A scalar field on a set $D \subseteq \mathbb{R}^n$ is a function $f \colon D \to \mathbb{R}$ , that is, an assignment of one real number to each point of $D$ . In this course $n$ is usually $2$ or $3$ . The word field stresses that the inputs are points of the plane or of space, often with a physical meaning such as temperature, pressure, density or electric potential.

In plain words A scalar field is a number attached to every point of a region. A weather map of temperatures is one.

Example Suppose the temperature in degrees Celsius at the point $(x, y, z)$ of a room, in meters from one corner, is

T(x, y, z) = 100 - x^2 - 2y^2 - 3z^2.

At the point $(1, 2, 1)$ the temperature is $100 - 1 - 8 - 3 = 88$ degrees.

Graphs

Definition The graph of a function $f$ of two variables with domain $D$ is the set of points in space

\begin{gathered} \{\, (x, y, z) \in \mathbb{R}^3 \mid \\ (x, y) \in D,\ z = f(x, y) \,\}. \end{gathered}

It is a surface sitting above or below the domain in the $xy$ -plane, with height $f(x, y)$ over each input point. A function of three variables has a graph in $\mathbb{R}^4$ , which cannot be drawn, so level surfaces are used instead.

In plain words Put the input point on the floor and raise a marker to the height of the output. All the markers together form the graph.

Example The graph of $f(x, y) = x^2 + y^2$ is the bowl-shaped surface $z = x^2 + y^2$ , called a paraboloid. The point $(1, 2, 5)$ lies on it because $1^2 + 2^2 = 5$ , and the point $(1, 2, 4)$ does not. The graph of the function $\sqrt{4 - x^2 - y^2}$ from the first section is the upper half of the sphere $x^2 + y^2 + z^2 = 4$ .

Traces

Definition A trace of a surface is its intersection with a plane. For the graph of $z = f(x, y)$ the most useful traces are the vertical ones. In the plane $x = a$ the trace is the curve $z = f(a, y)$ , and in the plane $y = b$ it is the curve $z = f(x, b)$ , each defined for the inputs that lie in the domain. Horizontal traces, in planes $z = c$ , are the level curves of the next section lifted to height $c$ .

In plain words Slice the surface with a flat sheet and look at the cut. Freezing one input turns the function into an ordinary function of the other input.

Example For the saddle $z = y^2 - x^2$ the trace in the plane $x = 0$ is the parabola $z = y^2$ , which opens upward. The trace in the plane $y = 0$ is the parabola $z = -x^2$ , which opens downward. In the plane $x = 1$ the trace is $z = y^2 - 1$ , the same upward parabola shifted down by $1$ . So the surface rises in the $y$ direction and falls in the $x$ direction, which is the shape of a saddle.

Level curves and contour maps

Definition For a function $f$ of two variables with domain $D$ and a real number $c$ , the level curve of $f$ at level $c$ is the set

\{\, (x, y) \in D \mid f(x, y) = c \,\}.

It lies in the $xy$ -plane and is empty when $c$ is not in the range of $f$ . A contour map is a drawing of several level curves, usually for equally spaced values of $c$ , each labeled with its value.

In plain words A level curve joins all the inputs that give the same output. A contour map is the hiking map version of the graph. When the levels are equally spaced, close curves mean steep ground and spread out curves mean gentle ground.

Example For $f(x, y) = x^2 + y^2$ the level curve at level $c$ is empty for $c < 0$ , is the single point $(0, 0)$ for $c = 0$ , and is the circle of radius $\sqrt{c}$ for $c > 0$ , passing through $(\sqrt{c}, 0)$ . Compare it with the cone $g(x, y) = \sqrt{x^2 + y^2}$ , whose level curve at level $c > 0$ is the circle of radius $c$ .

Level $c$	Radius for $x^2 + y^2$	Radius for $\sqrt{x^2 + y^2}$
$1$	$1$	$1$
$2$	$1.414$	$2$
$3$	$1.732$	$3$
$4$	$2$	$4$

Both contour maps are rings of circles. The cone’s rings are evenly spaced because its height grows at a constant rate. The paraboloid’s rings crowd together farther out because it gets steeper.

Level surfaces

Definition For a function $F$ of three variables with domain $D \subseteq \mathbb{R}^3$ and a real number $k$ , the level surface at level $k$ is

\{\, (x, y, z) \in D \mid F(x, y, z) = k \,\}.

In plain words These are all the points in space where the scalar field has one fixed value, such as all the points of a room at exactly 20 degrees.

Example Let $F(x, y, z) = x^2 + y^2 - z^2$ . For $k = 1$ the level surface $x^2 + y^2 - z^2 = 1$ is a hyperboloid of one sheet, and it contains $(1, 0, 0)$ . For $k = 0$ it is the double cone $z^2 = x^2 + y^2$ , which contains $(3, 4, 5)$ since $9 + 16 - 25 = 0$ . For $k = -1$ it is the hyperboloid of two sheets $z^2 - x^2 - y^2 = 1$ , which contains $(0, 0, 1)$ and $(0, 0, -1)$ but no point with $\lvert z \rvert < 1$ .

Interior and boundary points

Definition For a point $P$ of $\mathbb{R}^n$ and a number $r > 0$ , the open ball of radius $r$ about $P$ is the set of points $Q$ with $\lvert Q - P \rvert < r$ , an open disk when $n = 2$ . Let $S \subseteq \mathbb{R}^n$ . A point $P$ is an interior point of $S$ if some open ball about $P$ lies entirely in $S$ . A point $P$ is a boundary point of $S$ if every open ball about $P$ contains at least one point of $S$ and at least one point not in $S$ . The set of all boundary points is the boundary of $S$ . A boundary point may or may not belong to $S$ .

In plain words An interior point has some breathing room inside the set. A boundary point sits right at the edge, with the set and the outside both arbitrarily close.

Example Let $S$ be the open unit disk $x^2 + y^2 < 1$ . The point $(0.5, 0)$ is interior, because the disk of radius $0.5$ about it stays inside $S$ . The point $(1, 0)$ is a boundary point. Every open ball about $(1, 0)$ contains a smaller one of some radius $r$ with $0 < r < 1$ , and that smaller ball contains $(1 - r/2, 0)$ , which is in $S$ , and $(1 + r/2, 0)$ , which is not. The boundary of $S$ is the whole unit circle, and none of it belongs to $S$ .

Open and closed sets

Definition A set $S \subseteq \mathbb{R}^n$ is open if every point of $S$ is an interior point of $S$ , equivalently if $S$ contains none of its boundary points. A set is closed if it contains all of its boundary points, equivalently if its complement is open. Many sets are neither, and the whole space $\mathbb{R}^n$ and the empty set are both open and closed.

In plain words An open set leaves out its entire edge. A closed set includes its entire edge. A set that keeps part of its edge is neither.

Example The domain of $g(x, y) = \ln(y - x^2)$ is the region $y > x^2$ above a parabola. Its boundary is the parabola $y = x^2$ , and none of those points belong to the domain, so the domain is open. The domain of $\sqrt{4 - x^2 - y^2}$ is the closed disk $x^2 + y^2 \le 4$ , which contains its boundary circle, so it is closed. The strip $0 < x \le 1$ contains the boundary line $x = 1$ but not the boundary line $x = 0$ , so it is neither open nor closed.

Boundedness

Definition A set $S \subseteq \mathbb{R}^n$ is bounded if it lies inside some ball about the origin, that is, if there is a number $M$ with $\lvert P \rvert \le M$ for every point $P$ in $S$ . Otherwise $S$ is unbounded. A set that is both closed and bounded is called compact. Later you will see that a continuous function on a nonempty compact set always attains a largest and a smallest value.

In plain words A bounded set fits inside a big enough circle or sphere. An unbounded one runs off to infinity in some direction.

Example The closed disk $x^2 + y^2 \le 4$ is bounded, with $M = 2$ , and since it is also closed it is compact. The region $y > x^2$ is unbounded, because it contains every point $(0, n)$ with $n$ a positive integer, and those points have distance $n$ from the origin, which exceeds any proposed $M$ .

Check yourself

Find the domain and range of $f(x, y) = \ln(9 - x^2 - y^2)$ .

Show the answer

The logarithm needs $9 - x^2 - y^2 > 0$ , so the domain is the open disk $x^2 + y^2 < 9$ . On it the quantity $9 - x^2 - y^2$ takes every value in $(0, 9]$ , with $9$ at the origin. The logarithm of those values fills the range $(-\infty, \ln 9]$ .

Describe the level curves of $f(x, y) = y - x^2$ at the levels $c = -1$ , $c = 0$ and $c = 2$ .

Show the answer

Setting $y - x^2 = c$ gives the parabola $y = x^2 + c$ . The three level curves are the parabolas $y = x^2 - 1$ , $y = x^2$ and $y = x^2 + 2$ , which are copies of one parabola stacked vertically. The vertical gap between two of them equals the difference of their levels, so these three are not evenly spaced.

Is the ring $S = \{\, (x, y) \mid 1 \le x^2 + y^2 < 4 \,\}$ open, closed, bounded?

Show the answer

The boundary of $S$ is the two circles $x^2 + y^2 = 1$ and $x^2 + y^2 = 4$ . The set contains the inner circle, so it is not open, and it misses the outer circle, so it is not closed. It is bounded, since every point satisfies $\lvert P \rvert < 2$ .

Functions of Several Variables

Key terms

Domain and range

Scalar field

Graphs

Traces

Level curves and contour maps

Level surfaces

Interior and boundary points

Open and closed sets

Boundedness

Check yourself

How this lesson connects