Level sets and sections.

1. Jan 22, 2004

babbagee

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. Jan 23, 2004

HallsofIvy

Staff Emeritus
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".)