Level Curves and Contour Maps

12 min read

Textbook Reference

OpenStax Calculus Volume 3 is free and openly licensed; its third objective for this section is to sketch several traces or level curves of a function of two variables. The Stewart definition and worked examples cited below appear on pages 977-982.

Key idea

You have already read a contour map without thinking of it as mathematics. A hiking map shows curving lines, each labeled with an elevation, and you know that walking along one of those lines keeps you at the same height while crossing many of them quickly means a steep climb. That is exactly a level-curve diagram, and you have been interpreting one for years.

Here is the picture. A surface floats above the floor. Slice it with a horizontal plane at height $k$, like cutting a mountain with a sheet of glass laid flat. The slice is a curve on the surface, all at the same height $k$. Now let that curve drop straight down onto the floor, leaving its shadow. That shadow is the level curve $f(x, y) = k$: the set of floor points where the function equals $k$. Draw the shadows for several evenly spaced heights and you get a contour map, a flat drawing that records a three-dimensional surface. Where the shadows crowd together the surface is steep; where they spread apart it is gentle.

So a level curve is a surface’s shadow at one height, and a contour map is the flat record of the whole surface. This is how you read a curved shape using only a piece of paper.

Prerequisite Check

Before this lesson, make sure you can do all of the following:

• State what the graph of a function of two variables is (a surface above the floor)
• Recognize that $f(x, y) = k$ is an equation in $x$ and $y$ with $k$ held fixed
• Graph a line and a circle in the plane
• Understand a horizontal trace as a slice of the surface at constant height
• Read which side of a curve has larger function values

If the surface-and-trace picture is shaky, review Graphs of Functions and Regions, Lines, and Circles in the Plane.

Quick Reference

Definition. The level curves of a function $f$ of two variables are the curves with equation \[ f(x, y) = k, \] where $k$ is a constant in the range of $f$. A level curve is the set of all floor points at which $f$ takes the single value $k$.

A collection of level curves drawn for several values of $k$ is a contour map. The curves are usually drawn for equally spaced values of $k$.

The key relationship. The level curve $f(x, y) = k$ is the horizontal trace of the surface in the plane $z = k$, projected straight down to the floor. Lifting each level curve up to its height $k$ rebuilds the surface.

Reading steepness. Where level curves are close together, the surface is steep (height changes fast over a short distance). Where they are far apart, the surface is gentle.

Named contour maps in the real world. Equal-elevation curves on a topographic map; isothermals joining points of equal temperature; isobars joining points of equal atmospheric pressure.

The procedure for drawing level curves (flashcard size).

1. Set $f(x, y) = k$.
2. Choose several values of $k$ (equally spaced, inside the range).
3. For each $k$, identify the curve the equation describes (a line, a circle, an ellipse).
4. Draw all the curves on one floor, labeling each with its $k$ value.

Key Concepts

1. Estimating values from a contour map

A contour map lets you read off function values without a formula.

Example 1. A contour map for a function $f$ has labeled curves. To estimate $f(1, 3)$, locate the point $(1, 3)$ on the map. If it lies partway between the curves labeled $70$ and $80$, then \[ f(1, 3) \approx 73. \] The value is found by interpolating between the two nearest labeled curves. (Source: Stewart 14.1, Example 9, p. 980.)

Inline self-check. On a topographic map, you cross five closely spaced contour lines over a short horizontal distance, then walk a long way before crossing the next one. Where is the terrain steeper?

Show answer

Where the contour lines are close together. Crossing many equal-elevation lines over a short distance means the elevation changes quickly, which is a steep slope. The long gap means the elevation barely changes, which is gentle ground.

2. Level curves of a plane are parallel lines

Example 2. Sketch the level curves of $f(x, y) = 6 - 3x - 2y$ for $k = -6, 0, 6, 12$.

Goal. Set the function equal to each constant and identify the curves.

Step 1. Set $6 - 3x - 2y = k$, that is, $3x + 2y = 6 - k$.

Step 2. For each $k$ this is a line of slope $-\tfrac{3}{2}$:

• $k = -6$: $3x + 2y = 12$.
• $k = 0$: $3x + 2y = 6$.
• $k = 6$: $3x + 2y = 0$.
• $k = 12$: $3x + 2y = -6$.

Step 3. These are parallel lines, equally spaced because the $k$ values are equally spaced and the surface is a plane.

\[ \boxed{\text{The level curves are equally spaced parallel lines of slope } -\tfrac{3}{2}.} \]

Recap. A plane has straight, evenly spaced level curves; even spacing signals constant steepness, which matches a flat tilted surface. (Source: Stewart 14.1, Example 10, p. 980.)

3. Level curves of a hemisphere are concentric circles

Example 3. Sketch the level curves of $g(x, y) = \sqrt{9 - x^2 - y^2}$ for $k = 0, 1, 2, 3$.

Goal. Solve $g = k$ for the curve at each height.

Step 1. Set $\sqrt{9 - x^2 - y^2} = k$. Square: $9 - x^2 - y^2 = k^2$, that is, \[ x^2 + y^2 = 9 - k^2. \]

Step 2. For each $k$ this is a circle centered at the origin with radius $\sqrt{9 - k^2}$:

• $k = 0$: radius $3$.
• $k = 1$: radius $\sqrt{8} \approx 2.83$.
• $k = 2$: radius $\sqrt{5} \approx 2.24$.
• $k = 3$: radius $0$ (a single point at the origin).

Step 3. The level curves are concentric circles that shrink as the height $k$ rises, collapsing to a point at the top of the hemisphere.

\[ \boxed{\text{The level curves are concentric circles of radius } \sqrt{9 - k^2}.} \]

Recap. Lifting each circle to its height $k$ rebuilds the dome. Notice the circles are not evenly spaced even though the $k$ values are; the hemisphere flattens near the top and steepens near the rim, and the spacing of the circles shows exactly that. (Source: Stewart 14.1, Example 11, pp. 980-981.)

4. Level curves of a paraboloid are ellipses

For $h(x, y) = 4x^2 + y^2 + 1$, setting $4x^2 + y^2 + 1 = k$ gives $4x^2 + y^2 = k - 1$, which for $k > 1$ is an ellipse. The contour map is a family of nested ellipses growing outward as $k$ increases, and lifting them to their heights rebuilds the elliptic paraboloid bowl. A contour map is often more useful than a three-dimensional sketch for reading off exact values, which is why weather services and economists publish contour maps rather than perspective drawings. (Source: Stewart 14.1, Example 12 and Example 13, pp. 981-982.)

Recap. The shape of the level curves names the surface: parallel lines mean a plane, concentric circles mean a surface of revolution, nested ellipses mean an elliptic paraboloid.

Common Errors Summary

Common Misconceptions

Leveled Practice

Work each problem fully before revealing the answer.

Level 1 -- Direct Application

Problem 1. Sketch the level curves of $f(x, y) = x + y$ for $k = 0, 1, 2$.

Problem 2. What are the level curves of $f(x, y) = x^2 + y^2$ for $k = 1, 4, 9$?

Problem 3. A point on a contour map sits exactly on the curve labeled $50$. What is the function value there?

Level 2 -- Identifying Surfaces from Contours

Problem 4. Sketch the level curves of $g(x, y) = \sqrt{16 - x^2 - y^2}$ for $k = 0, 2, 4$.

Problem 5. The level curves of a function are parallel straight lines, equally spaced. What can you conclude about the surface?

Level 3 -- Reading and Interpreting Contour Maps

Problem 6. Two contour maps are given. On the first, the circles are equally spaced; on the second, the circles crowd together toward the center. One surface is a cone and the other is a paraboloid. Which is which, and why?

Mastery Checklist

You have mastered this skill when you can, without notes:

• State the definition of a level curve $f(x, y) = k$
• Explain a level curve as a horizontal trace projected down to the floor
• Draw the level curves of a plane (parallel lines) and of a surface of revolution (concentric circles)
• Estimate a function value by interpolating between labeled contours
• Read steepness from contour spacing (close means steep, far means gentle)
• Choose $k$ values inside the range of the function
• Identify a surface (plane, cone, paraboloid) from the shape and spacing of its level curves

Mental Model

Think of a contour map as a stack of horizontal slices flattened onto one page.

Imagine slicing a hill with a stack of glass sheets, one every few feet of elevation. Each sheet cuts the hill in a curve at one height. Now press all the sheets straight down onto the ground, keeping each curve where it landed and writing its elevation beside it. The page of nested, labeled curves is the contour map, a faithful flat record of the hill.

Reading the map runs the slicing backward. A labeled curve tells you the elevation everywhere along it. Curves packed tightly mean the hill rises steeply there, because you climb through many elevations over little ground. Curves spread thin mean gentle slope. Concentric closed curves around a single point mark a peak or a pit; parallel straight curves mark a uniform incline. The whole three-dimensional shape is recoverable from this flat drawing, which is why a single sheet of paper can carry a mountain.

Connections

Built from

• Graphs of Functions: A level curve is a horizontal trace of the graph, projected to the floor, so the graph and its contour map are two views of one surface.
• Regions, Lines, and Circles in the Plane: Each level curve is a line, circle, or ellipse drawn in the plane.

Leads to

• Functions of Three or More Variables: The same idea one dimension up gives level surfaces $f(x, y, z) = k$, the way a four-dimensional graph is made visible.
• The Gradient Vector: The gradient at a point is perpendicular to the level curve through that point and points in the direction of steepest increase, which is the calculus reading of “where the contours crowd.”

Where this shows up

Level curves are how a surface is read in two dimensions, and they connect directly to the most important idea in the chapter: the gradient points across the level curves toward steepest ascent, and its size is large exactly where the contours crowd. In Chapter 15, polar and other coordinate choices are guided by the shape of the level curves of the integrand. Outside mathematics, contour maps are the working language of topography, of weather (isothermals and isobars), of medical imaging, and of any field that visualizes a quantity over a two-dimensional region. Learning to read steepness and shape from a contour map is a broadly useful skill.

Audience Notes

For students who find math intimidating: You have read a hiking or weather map before. That is all a level-curve diagram is. The only new step is producing the curves from a formula by setting $f(x, y) = k$ and solving, which gives a line or a circle you already know how to draw.

For students who want depth: The level set $\{f = k\}$ is, for a smooth function with nonzero gradient, a smooth curve, and the Implicit Function Theorem guarantees you can locally describe it as $y$ as a function of $x$. The gradient is everywhere normal to the level set, which is the geometric seed of the method of Lagrange multipliers at the end of the chapter, where an extremum on a constraint curve occurs exactly where two families of level curves are tangent.

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