# 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 \(\mathbb{R}^n\) and range is \(\mathbb{R}\). For example

\[ f(x,y) = x - y \]

is a function of two variables

\[ g(z,y,x) = \dfrac{ x - y}{ y-z}\]

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 \(\PageIndex{1}\): Domain of a Function

Find the domain of

\[ f(x,y) = \dfrac{\sqrt{x-y}}{x+y}.\]

**Solution**

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

\[ x - y \ge 0\]

second, the denominator must be nonzero, that is

\[ x + y \neq 0\]

hence we need to stay off the line

\[y = -x.\]

Putting this together gives

\[(x,y) | x - y \ge 0 \;\; \text{ and } \;\; y \neq -x.\]

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

Exercise 1

Find the domain of the function

\[ f(x,y,z) = \dfrac{xyz}{\sqrt{4-x^2-y^2-z^2}}.\]

### 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.

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:

\[ f(x,y) = y - x^2\]

by setting constant values for \(z\):

\(z\) | Equation |

1 | \(y = x^2 + 1\) |

2 | \(y = x^2 + 2\) |

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.

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.