Drawing a contour map for a function $z = f(x, y)$ involves visualizing its three-dimensional graph as a series of level curves projected onto a two-dimensional plane.
Understanding Contour Maps
A contour map, also known as a level set or level curve map, provides a two-dimensional representation of a three-dimensional surface. It is essentially a topographical map of the function's graph. Each curve on the map connects points where the function's value ($z$) is constant.
Think of it like slicing the 3D graph of $z = f(x, y)$ with horizontal planes at different heights (values of $z$), and then looking down on the $xy$-plane to see where those slices intersect the graph.
Steps to Draw a Contour Map
Based on the process of creating a topographical map of the graph, here are the key steps:
- Choose Elevations (Level Values): You start by selecting a set of equally spaced constant values for $z$. Let these values be $c_1, c_2, c_3, \dots, c_n$. The choice of spacing depends on the desired detail; closer spacing reveals more subtle changes in elevation.
- Find the Level Curves: For each chosen elevation $c_i$, you find the set of all $(x, y)$ points in the domain of the function such that $f(x, y) = c_i$. This equation, $f(x, y) = c_i$, defines a curve (or possibly multiple curves or even points) in the $xy$-plane. These are the level curves corresponding to the elevation $c_i$.
- Reference Insight: This step involves finding points on the graph for each elevation $z=c$.
- Project onto the xy-Plane: The curves found in step 2, which lie on the graph at a specific height $z=c_i$, are then projected down onto the $xy$-plane. Since the equation $f(x, y) = c_i$ is already an equation in terms of $x$ and $y$, solving or identifying these curves directly gives you their projection onto the $xy$-plane.
- Reference Insight: You project the curves on the graph onto the xy-plane.
- Plot the Level Curves: Draw all the level curves obtained in step 3 on a single $xy$-plane. Label each curve with its corresponding $z$-value ($c_i$).
Practical Considerations
- Spacing of Levels: Choosing levels that are too far apart might miss important features like peaks or valleys. Choosing them too close might make the map cluttered.
- Identifying Curves: Depending on the function $f(x, y)$, the equation $f(x, y) = c$ might represent simple shapes like circles, lines, or parabolas, or more complex curves that require plotting points.
- Domain: Only consider points $(x, y)$ that are within the domain of the function $f$.
- Visual Interpretation: Closely spaced contour lines indicate a steep slope on the function's graph, while widely spaced lines indicate a gentle slope.
Example: $f(x, y) = x^2 + y^2$
Let's sketch a contour map for the function $z = f(x, y) = x^2 + y^2$.
- Choose Levels: Let's pick simple, equally spaced positive $z$-values: $c = 0, 1, 4, 9$. (Choosing positive values is logical since $x^2 + y^2$ is always non-negative).
- Find Level Curves:
- For $z = 0$: $x^2 + y^2 = 0$. The only solution is $(x, y) = (0, 0)$. This is a single point at the origin.
- For $z = 1$: $x^2 + y^2 = 1$. This is the equation of a circle centered at the origin with radius $\sqrt{1} = 1$.
- For $z = 4$: $x^2 + y^2 = 4$. This is the equation of a circle centered at the origin with radius $\sqrt{4} = 2$.
- For $z = 9$: $x^2 + y^2 = 9$. This is the equation of a circle centered at the origin with radius $\sqrt{9} = 3$.
- Project onto xy-Plane: The equations $x^2+y^2=c$ are already defined in the $xy$-plane.
- Plot: Plot these curves on the $xy$-plane. You will see concentric circles centered at the origin, labeled with their corresponding $z$-values (0 at the center, then 1, 4, 9 outwards).
This process of setting $f(x, y) = c$ and plotting the resulting curves for various values of $c$ is the standard method for drawing a contour map.
