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M408M Learning Module Pages
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Chapter 10: Parametric Equations and Polar Coordinates

Chapter 12: Vectors and the Geometry of Space

Learning module LM 12.1: 3-dimensional rectangular coordinates:

Learning module LM 12.2: Vectors:

Learning module LM 12.3: Dot products:

Learning module LM 12.4: Cross products:

Learning module LM 12.5: Equations of Lines and Planes:

Learning module LM 12.6: Surfaces:

      Surfaces and traces
      Level curves
      Level surfaces
      Worked problems

Chapter 13: Vector Functions


Chapter 14: Partial Derivatives


Chapter 15: Multiple Integrals



Level curves

Level Curves
For a general function z=f(x,y), slicing horizontally is a particularly important idea:

Level curves: for a function z=f(x,y):DR2R the level curve of value c is the curve C in DR2 on which f|C=c.

Notice the critical difference between a level curve C of value c and the trace on the plane z=c: a level curve C always lies in the xy-plane, and is the set C of points in the xy-plane on which f(x,y)=c, whereas the trace lies in the plane z=c, and is the set of points (x,y,c) with (x,y) in C.

    By combining the level curves f(x,y)=c for equally spaced values of c into one figure, say c=1,0,1,2,, in the xy-plane, we obtain a contour map of the graph of z=f(x,y). Thus the graph of z=f(x,y) can be visualized in two ways,

  one as a surface in 3-space, the graph of z=f(x,y),

  the other as a contour map in the xy-plane, the level curves of value c for equally spaced values of c.
As we shall see, both capture the properties of z=f(x,y) from different but illuminating points of view. The particular cases of a hyperbolic paraboloid and a paraboloid are shown interactively in

Problem: Describe the contour map of a plane in 3-space.
Solution: The equation of a plane in 3-space is Ax+By+Cz = D, so the horizontal plane z=c intersects the plane when Ax+By+Cc = D.
For each c, this is a line with slope A/B and y-intercept y=(DCc)/B. Since the slope does not depend on c, the level curves are parallel lines, and as c runs over equally spaced values these lines will be a constant distance apart.

Consequently, the contour map of a plane consists of equally spaced parallel lines.
(Does this make good geometric sense?)