Traces
3
TOPIC PAGE: MULTIVARIABLE CALCULUS TRACES, LEVEL CURVES, AND CONTUOUR MAPS Click here for a printable .pdf version TRACES The trace of a surface in a plane is the intersection of the surface with that plane. While we can discuss traces in any plane, for surfaces in the form z = f(x,y) we are particularly interested in traces in planes parallel to the xy plane. Note that the xy plane has the equation z = 0, and planes parallel to the xy plane have equations in the form z = a. To obtain the trace of z=f(x,y) in the plane z=a, we set f(x,y) = a This implicitly defines a 2D curve lying in the plane z = a. LEVEL CURVES The level curves (or contour lines) of a surface are paths along which the values of z = f(x,y) are constant; i.e. the level curves of a function are simply the traces of that function in various planes z = a, projected onto the xy plane. The example shown below is the surface Examine the level curves of the function. Sliding the slider will vary a from a = -1 to a = 1. On the left, you'll see the planes z = a pass through the surface; the intersections are the traces. On the right, you'll see the level curve associated with each value of a. The slider occasionally sticks. If this happens, leave the page and return to it. Even better, just use the play button to advance through the values of a. CONTOUR MAPS On the left, we have a three dimensional contour plot of the surface shown above; instead of showing the entire surface, the plot shows curves along the surface corresponding to traces in the planes z = a. On the right, we have a contour map of of the surface, showing level curves of the function in the xy plane (we're taking all the level curves revealed by the sliding a value, and plotting them simultaneously). The trace in the plane z = 1 is the ellipse The trace in the plane z = 3 is the ellipse
description
traces
Transcript of Traces
TOPIC PAGE: MULTIVARIABLE CALCULUS
TRACES, LEVEL CURVES, AND CONTUOUR MAPS
Click here for a printable .pdf version TRACES
The trace of a surface in a plane is the intersection of the surface with that plane. While we can discuss traces in any plane, for surfaces in the form z = f(x,y) we are particularly interested in traces in planes parallel to the xy plane. Note that the xy plane has the equation z = 0, and planes parallel to the xy plane have equations in the form z = a. To obtain the trace of z=f(x,y) in the plane z=a, we set
f(x,y) = a
This implicitly defines a 2D curve lying in the plane z = a.
LEVEL CURVES
The level curves (or contour lines) of a surface are paths along which the values of z = f(x,y) are constant; i.e. the level curves of a function are simply the traces of that function in various planes z = a, projected onto the xy plane.
The example shown below is the surface
Examine the level curves of the function. Sliding the slider will vary a from a = -1 to a = 1. On the left, you'll see the planes z = a pass through the surface; the intersections are the traces. On the right, you'll see the level curve associated with each value of a.
The slider occasionally sticks. If this happens, leave the page and return to it. Even better, just use the play button to advance through the values of a.
CONTOUR MAPS
On the left, we have a three dimensional contour plot of the surface shown above; instead of showing the entire surface, the plot shows curves along the surface corresponding to traces in the planes z = a. On the right, we have a contour map of of the surface, showing level curves of the function in the xy plane (we're taking all the level curves revealed by the sliding a value, and plotting them simultaneously).
The trace in the plane z = 1 is the ellipse
The trace in the plane z = 3 is the ellipse
In the contour map above, I've maintained the orientation of the x and y axes to match the orientation of the 3D surface. I'm going to rotate it around to the more conventional xy orientation, remove a few of the contour lines (it's kind of crowded!), and make some observations:
Takes a bit of practice to read these and imagine the colors or values translated into heights, but contour maps are used for a bunch of things - here's a topographic map (which is a contour map showing heights of terrain).
A contour map should give some indication about the shape of the surface it is representing. This one has a problem - once you flatten out the surface by projecting the level curves onto the xy plane, you lose information about the height of the function; we know that the height of the function is constant along each curve, but we don't know what that height is.
There are (at least) two ways to indicate height information on a contour map:
1. through the use of color
2. by adding numerical data to the curves
These are shown in the images below.
Colored surface with contours 3D colored contours 2D colored contour map Legend
Contour map with numerical data
For a bigger picture, go here:
http://vulcan.wr.usgs.gov/Volcanoes/CraterLake/Maps/map_crater_lake_topo.html
EXERCISE:
Let
Write the equations that give the level curves for a = 0, 1, 2, 3, 4, 5. Describe these curves, and sketch a contour map that includes numerical data. Try to construct a 3D representation of the image based on the contour map.
