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1.2: Functions of Several Variables

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A function of several variables is a function where the domain is a subset of Rn and range is R. For example

f(x,y)=xy

is a function of two variables

g(z,y,x)=xyyz

is a function of three variables.

Finding the Domain

To find the domain of a function of several variables, we look for zero denominators and negatives under square roots:

Example 1.2.1: Domain of a Function

Find the domain of

f(x,y)=xyx+y.

Solution

First, the inside of the square root must be positive (since we are discussing real numbers), that is

xy0

second, the denominator must be nonzero, that is

x+y0

hence we need to stay off the line

y=x.

Putting this together gives

(x,y)|xy0 and yx.

The graph to the right shows the domain as the shaded green region.

alt

Exercise 1

Find the domain of the function

f(x,y,z)=xyz4x2y2z2.

Contours (Level Curves)

The topographical map shown below is of the Rubicon Trail. It represents the function that maps a longitude and latitude to an altitude.

alt

Each curve represents a path where the z-coordinate (altitude) is a constant. Crossing many topo lines in a short distance represents a path that is very steep. Now lets make our own contour map of the function:

f(x,y)=yx2

by setting constant values for z:

z Equation
1 y=x2+1
2 y=x2+2

We see that each topo line is a parabola and that the y-intercept gives the height. Below is a contour diagram of this function.

alt

Names for the curves drawn are level curves, isotherms (for temperature), isobars (for pressure), and equipotential lines (for electric potential fields) depending on what the two variable function represents.

Larry Green (Lake Tahoe Community College)

  • Integrated by Justin Marshall.


This page titled 1.2: Functions of Several Variables is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Larry Green.

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