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Level sets and sections.

  1. Jan 22, 2004 #1
    I am having trouble with these problems.

    Describe the graph of each function by computing some level sets and sections.

    f:R3-->R,(x,y,z)|-->xy

    the part that i am having trouble with is R3-->R what does that mean. As for computing the level sets all i do is set xy=to some constant. And then sketch it on the 3d plane.

    Thanks.
     
  2. jcsd
  3. Jan 23, 2004 #2

    HallsofIvy

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    R3->R simply means that the independent variable is from R3 (in other words (x,y,z)) while the value of the function is in R (a single real number).
    In this particular case f(x,y,z)= xy. Since there are 4 variables, x, y, z, and the function value, f, in order to "graph" it you would need a 4-dimensional drawing. What you can, in general, do is draw, in 3-dimensions, f(x,y,z)= C for different values of C. You could then interpret the 4th variable as time and imagine the drawings as pages in a flip book (frames in an animation for more modern people).

    For example, taking C= 1, xy= C is a hyperbola (strictly speaking a hyperbolic cylinder with axis parallel to the z axis). xy= -1, xy= 2, xy= -2, etc. give different hyperbolic cylinders showing how the system "evolves over time".

    This particular problem was probably created to be particularly easy. Since there is no z in the formula, you can actually draw them in a 2 dimensional xy- graph and imagine them extending into and out of the plane of the graph.

    (There is, by the way, no such thing as "the 3d plane". Planes are, by definition, 2 dimensional. You probably meant "in 3 dimensions".)
     
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