Math 314 Lecture #11 14.1: Functions of Several Variables

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.

Example. Let z = f (x, y) = 25 - x2 - y2.

The domain of this is D = {(x, y) : x2 + y2 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 + y2 = 25.

The largest value f realizes is 25 = 5; this occurs when x2 + y2 = 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 z2 = 25 - x2 - y2, i.e., x2 + y2 = 25 - z2.

The range of f is the closed interval [0, 5].

Example. Let w = f(x, y, z) =

1 .

4 - x2 - y2 - z2

The domain is D = {(x, y, z) : 4 - x2 - y2 - z2 > 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 + y2 + z2 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 - y2.

The domain is the whole of the xy-plane, and the range is the interval (-, 3]. We recognize the equation z = 3 - x2 - y2 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) = xy2 - x3 is the whole xy-plane and the range

is the whole real line. The level curves of f are the curves xy2 - x3 = k, i.e., y = ? (k + x3)/x when x = 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 + 3y2 + 5z2 is the set of

nonnegative real numbers. The level surfaces of f are the ellipsoids x2 + 3y2 + 5z2 = 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.

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