Math 314 Lecture #11
§14.1: Functions of Several Variables
A function of two variables is a rule f that assigns to each ordered pair of real numbers
(x, y) in a set D a unique number f (x, y).
The domain of f is the set D if specified, and is otherwise the set D of points (x, y) for
which the rule f makes sense.
The range of f is the set of real numbers that f realizes, i.e., {f (x, y) : (x, y) ∈ D}.
We often write z = f (x, y), and we call x and y the independent variables and z the
dependent variable.
Outcome A: Find and sketch (by hand) the domain of a function of two or more variables.
√
Example. Let z = f (x, y) = x + y.
Since no domain was specified, we seek for the set of points (x, y) for which the function
makes sense.
The square root requires that x + y ≥ 0; the domain is D = {(x, y) : x + y ≥ 0}. Here is
a rendering of this domain (the purple or shaded region) that includes the line x + y = 0.
√
Example. Let z = f (x, y) = y − x ln(y + x).
The square root requires that y −x ≥ 0 and the natural logarithm requires that y +x > 0.
The domain of f is that part of the xy-plane that satisfies both of these inequalities: that
is D = {(x, y) : y − x ≥ 0 and y + x > 0}. Here is a rendering of this domain (the purple
or shaded part) that includes part of the line y = x but excludes the line y = −x.
Outcome B: Find the range of a function of two or more variables.
p
Example. Let z = f (x, y) = 25 − x2 − y 2 .
The domain of this is D = {(x, y) : x2 + y 2 ≤ 25}, i.e., all of the points on or inside the
circle of radius 5 and center at the origin.
The smallest value f realizes is 0; this occurs when x2 + y 2 = 25.
√
The largest value f realizes is 25 = 5; this occurs when x2 + y 2 = 0.
The function realizes every value between 0 and 5 for appropriate choices of (x, y); for
0 ≤ z ≤ 5 we choose (x, y) in D such that z 2 = 25 − x2 − y 2 , i.e., x2 + y 2 = 25 − z 2 .
The range of f is the closed interval [0, 5].
1
Example. Let w = f (x, y, z) = p
.
4 − x2 − y 2 − z 2
The domain is D = {(x, y, z) : 4 − x2 − y 2 − z 2 > 0}, i.e., all of the points inside the
sphere of radius 2 with center at the origin, but not the points on this sphere.
Since we can choose points (x, y, z) in D for which x2 + y 2 + z 2 is close to 4, there are
arbitrarily large values in the range of f .
√
The smallest value that f realizes is f (0, 0, 0) = 1/ 4 = 1/2.
The function f realizes any value bigger than 1/2 for choices of (x, y, z) ∈ D.
The range of f is the interval [1/2, ∞).
Outcome C: Sketch (by hand) the graph of a function of two variables.
The graph of a function z = f (x, y) of two variables with domain D is the set of points
(x, y, z) in R3 such that z = f (x, y) with (x, y) ∈ D.
For a simple enough function, its graph might be a plane, a cylinder, or more generally,
a quadric surface.
Example. Let z = f (x, y) = 3 − x2 − y 2 .
The domain is the whole of the xy-plane, and the range is the interval (−∞, 3].
We recognize the equation z = 3 − x2 − y 2 as a quadric surface. It is the equation for a
circular paraboloid that opens downward with its peak at the point (0, 0, 3).
Outcome D: Identify the level curves and sketch (by hand) the contour map of a function
of two variables.
When the graph of a function is not a quadric surface, then we need to extract additional
information from the function to sketch its graph.
The level curves of a function f of two variables are the curves with equations f (x, y) =
k lying in the domain of f , where k is a constant in the range of f .
The level curves are just the horizontal traces of the graph of f .
Example. The function z = f (x, y) = x3 − y has as its domain D the whole xy-plane
and as its range the whole real line.
The level curves of f are the curves x3 − y = k lying in the xy-plane for any value of k.
These level curves are the graphs of the cubic function y = x3 − k. Here is a sampling
of these level curves for k = −3, −2, −1, 0, 1, 2, 3. Which of these curves corresponds to
larger values of k? smaller values of k?
Here is the graph of this function of two variables.
Example. The domain of z = f (x, y) = xy 2 − x3 is the whole xy-plane and the range
is the whole real line.
p
The level curves of f are the curves xy 2 − x3 = k, i.e., y = ± (k + x3 )/x when x 6= 0.
Here is a sampling of these level curves. Which curves corresponding to larger values of
k? to smaller values of k?
Here is the graph of this function. This graph is called a “monkey saddle” as it provides
places for the monkey’s legs and tail.
Outcome E: Describe the level surfaces of a function of three variables.
The graph of a function w = f (x, y, z) of three variables lies in 4-dimensional space, and
so we will not attempt to render its graph.
The level surfaces of w = f (x, y, z) are the surfaces determined by the equation
f (x, y, z) = k for values of k in the range of f .
Example. The range of the function w = f (x, y, z) = x2 + 3y 2 + 5z 2 is the set of
nonnegative real numbers.
The level surfaces of f are the ellipsoids x2 + 3y 2 + 5z 2 = k for k > 0, and the single
point (0, 0, 0) when k = 0.
If we think of w = k ≥ 0 as “time,” then we can imagine the graph of f as an animation of
its levels surfaces for increasing value of k: it starts with the point (0, 0, 0), then becomes
an ever increasing ellipsoid.