Math 2415 – Calculus IIISection 14.1 Functions of Several Variables

• Often, a quantity may depend on more than one input value. For example, temperature T on the surface of theearth at any given time t, where t is fixed, depends on latitude and longitude. => T is a function of x and y.T = f (x,y)

• Can also think of the volume of a circular cylinder as a function of two variables: V (r,h) = πr2h

• Definition: A function f of two variables is a rule that assigns to each ordered pair of real numbers (x,y) in aset D a unique number denoted by f (x,y). The domain D is a subset of the xy-plane and the range is the set off (x,y).

• Write z = f (x,y) where is the dependent variable and are independent variables.

• NOTE: If a function f is given by a formula and no domain specified then the domain is the set of all (x,y)such that the expression is well-defined.

Ex: Find the domains of the following functions and evaluate f (3,2).

a). f (x,y) =√

x+ y+1x−1

b). f (x,y) = x ln(y2− x)

• Read example 2 – wind chill depends on temperature and wind speed.Ex: Find the domain and range of g(x,y) =

√49− x2− y2

Math 2415 Section 14.1 Continued

• If f is a function of two variables with domain D then the graph of f is the set of all points (x,y,z) insuch that

• NOTE: The graph of a function f of one variable is a curve C (y = f (x)) The graph of a function f of twovariables is a surface S (z = f (x,y)). We can visualize the graph S of f as lying directly above or below itsdomain in the xy plane. See Figure 5.

Ex: Sketch the graph of the function z = f (x,y) = 8−4x−2y

Ex: Sketch the graph of z = g(x,y) =√

16− x2− y2

Ex: Find the domain and range and sketch the graph of z = h(x,y) = 4x2 + y2.

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Math 2415 Section 14.1 Continued

• Level curves The level curves of a function f of two variables are the curves with equationswhere

• Level curves lie in the . They are the in the of thein the plane

• Visualize level curves being lifted to the surface at the indicated height.

• If level curves are close together =>

• If level curves are farther apart =>

•

• Some examples of level curves are topographic maps – level curves givetemperature functions – level curves are isothermals – join

Ex: Given a contour map for f , estimate the values of f (1,3) and f (4,5).

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Math 2415 Section 14.1 Continued

Ex: Sketch some level curves of the function f (x,y) = x+ y+2.

Ex: Sketch some level curves of the function g(x,y) =√

25− x2− y2

Ex: Sketch some level curves of the function h(x,y) = x2 +9y2.

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Math 2415 Section 14.1 Continued

•Functions of three or more variables

Ex: Find the domain of f if f (x,y,z) = ln(z− y)− zcos(xy)

• It is difficult to visualize a function of 3 or more variables => requires 4D or higher.

• We can look at level surfaces to visualize functions of 3 variables. These have equation f (x,y,z) = k. If apoint (x,y,z) moves along a level surface, the value of f (x,y,z) is

Ex: Describe the level surfaces of the function f (x,y,z) = x2 + y2 + z2.

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