Section 4.1 - Functions of Several Variables

Section 4.1 - Functions of Several Variables

Functions of Two Variables

In multivariable calculus, a function of two variables assigns a real number to each ordered pair of real numbers $(x, y)$ in a set $D$. The set $D$ is called the domain of the function, and the set of values the function takes is called its range.

• Definition: A function $f$ of two variables is a rule that assigns to each $(x, y)$ in a set $D$ a unique real number, denoted by $f(x, y)$.

• Domain ($D$): The set of all possible input pairs $(x, y)$ for which $f(x, y)$ is defined.

• Range: The set of all possible output values $f(x, y)$ as $(x, y)$ varies over $D$.

• Notation: $f: D \to \mathbb{R}$

We often write $z = f(x, y)$ to make explicit the value taken by $f$ at the general point $(x, y)$. The variables $x$ and $y$ are independent, and $z$ is the dependent variable.

Examples of Functions of Two Variables

• Example 1: For each of the following functions, evaluate $f(3,2)$, find and sketch the domain.

  • $f(x, y) = \frac{x^2 + y^2}{x - 1}$

  • $g(x, y) = \ln(y^2 - x)$

• Example 2: Find the domain and range of $f(x, y) = \sqrt{9 - x^2 - y^2}$.

Additional info: To find the domain of $f(x, y) = \sqrt{9 - x^2 - y^2}$, set $9 - x^2 - y^2 \geq 0$, so the domain is the disk $x^2 + y^2 \leq 9$. The range is $0 \leq f(x, y) \leq 3$.

Graphs of Functions of Two Variables

The graph of a function $f(x, y)$ of two variables is the set of all points $(x, y, z)$ in three-dimensional space such that $z = f(x, y)$.

• A function of the form $f(x, y) = ax + by + c$ is called a linear function. The graph of such a function is a plane with the equation $ax + by - z + c = 0$.

• Example 4: Sketch the graph of $g(x, y) = \sqrt{9 - x^2 - y^2}$.

Additional info: The graph of $f(x, y) = \sqrt{9 - x^2 - y^2}$ is the upper hemisphere of a sphere of radius 3 centered at the origin.

Level Curves

Level curves (also called contour curves) of a function $f$ of two variables are the curves with equations $f(x, y) = k$, where $k$ is a constant in the range of $f$.

• Level curves help visualize the function by showing where it takes constant values.

• For example, the level curves of $f(x, y)$ are the set of points $(x, y)$ where $f(x, y)$ is constant.

Example: The image shows the level curves of a function $f(x, y)$, which are often used in topographical maps and other applications to represent regions of equal value.

Term	Definition	Example
Function of two variables	Rule assigning a real number to each $(x, y)$ in a set $D$	$f(x, y) = x^2 + y^2$
Domain	Set of all $(x, y)$ for which $f(x, y)$ is defined	For $f(x, y) = \sqrt{9 - x^2 - y^2}$, $x^2 + y^2 \leq 9$
Range	Set of all possible values of $f(x, y)$	For $f(x, y) = \sqrt{9 - x^2 - y^2}$, $0 \leq f(x, y) \leq 3$
Level curve	Curve where $f(x, y) = k$ for constant $k$	For $f(x, y) = x^2 + y^2$, level curves are circles $x^2 + y^2 = k$

Additional info: Level curves are essential for visualizing functions of two variables and are widely used in fields such as physics, engineering, and geography.