Functions of Several Variables: Graphs, Surfaces, and Contour Diagrams

Chapter 12: Functions of Several Variables

12.1 Functions of Two Variables

In calculus, a function of two variables is a rule that describes how one quantity depends on two others. This extends the concept of single-variable functions to higher dimensions, allowing us to model more complex relationships.

• Definition: A function of two variables, $f(x, y)$, assigns a real number to each pair $(x, y)$ in its domain.

• Examples:

  • The area of a rectangle is a function of its length and width: $A = l \times w$.

  • BMI (Body Mass Index) is a function of height and weight.

Height (in)	Weight (lb)	BMI
60	120	23.4
62	140	25.6
64	160	27.5
66	180	29.0
68	200	30.4

Additional info: Table shows how BMI depends on both height and weight.

• 3-D Coordinate System: To graph functions of two variables, we use three axes: $x$, $y$, and $z$.

• Right-handed axes: The $x$-axis is usually horizontal, $y$-axis is perpendicular, and $z$-axis is vertical.

• Plotting in 3D: Points are represented as $(x, y, z)$, e.g., $(1, 2, 3)$.

12.1.1 Graphs of Planes in 3D

Equations like $z = 0$, $z = 3$, $z = -1$ represent planes in three-dimensional space.

• $z = 0$: The $xy$-plane.

• $z = 3$: A plane parallel to the $xy$-plane, three units above.

• $z = -1$: A plane parallel to the $xy$-plane, one unit below.

Example: Which point is closest to the $xz$-plane? The point with $y$-value closest to zero.

• Distance Formula in 2D: For points $(x_1, y_1)$ and $(x_2, y_2)$: $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

• Distance Formula in 3D: For points $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$: $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$

12.2 Graphs & Surfaces

Functions of two variables can be visualized as surfaces in three-dimensional space. Common examples include paraboloids and saddle surfaces.

• Example Functions:

  • $f(x, y) = x^2 + y^2$ (paraboloid)

  • $g(x, y) = x^2 - y^2$ (saddle surface)

  • $K(x, y) = x^2 - (y - 1)^2$

• Cross-sections: By fixing one variable and letting the other vary, we obtain cross-sections of the surface.

  • For $f(x, y) = x^2 + y^2$, fixing $x = 1$ gives $f(1, y) = 1 + y^2$ (a parabola in $y$).

  • For $g(x, y) = x^2 - y^2$, fixing $x = 1$ gives $g(1, y) = 1 - y^2$ (an upside-down parabola in $y$).

• Saddle Surface: The graph of $g(x, y) = x^2 - y^2$ is called a saddle surface due to its shape.

12.3 Contour Diagrams

Contour diagrams (also called level curves) are used to represent functions of two variables by showing curves of constant value. These are analogous to topographic maps, where contour lines indicate points of equal elevation.

• Definition: For a function $f(x, y)$, the contour at height $c$ is the set of points where $f(x, y) = c$.

• Interpretation: Contour lines outline the shape of the surface and separate regions of different values.

• Applications: Used in geography (topographic maps), meteorology (isobars), and engineering.

Feature	Description
Peak	Highest point in the region
Ridge	Long, narrow elevation
Cliff	Steep slope
Valley	Low area between hills
Saddle	Low point between two peaks

• Example: For $f(x, y) = x^2 + y^2$, contours are circles centered at the origin: $x^2 + y^2 = c$ Radius of each contour: $\sqrt{c}$

• Example: For $f(x, y) = kx^2 + y^2$, contours are ellipses: $kx^2 + y^2 = c$

• Example: For $f(x, y) = 2x + 3y + 1$, contours are straight lines: $2x + 3y + 1 = c$

Additional info: Contour lines are obtained by taking horizontal slices of the surface at different heights.

Summary Table: Types of Contour Curves

Function	Contour Equation	Shape
$x^2 + y^2$	$x^2 + y^2 = c$	Circle
$kx^2 + y^2$	$kx^2 + y^2 = c$	Ellipse
$2x + 3y + 1$	$2x + 3y + 1 = c$	Straight line

Key Terms

• Function of several variables: A rule assigning a value to each point in a multi-dimensional domain.

• Surface: The graph of a function of two variables in 3D space.

• Contour line/level curve: Curve along which the function has constant value.

• Cross-section: A slice of the surface at a fixed value of one variable.

Applications

• Modeling physical phenomena (temperature, pressure, elevation)

• Visualizing solutions to equations in engineering and science

• Analyzing optimization problems with multiple variables