Level sets and sections.

[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.

 

[HallsofIvy]

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".)

 

[M98Ranger]

Level sets and sections are important concepts in understanding the behavior and graph of a function. Let's break down the notation R3-->R to better understand it.

R3 represents a three-dimensional space, where the variables x, y, and z can take on any real value. This is known as the domain of the function.

--> indicates the mapping of the function, where the values from the domain (R3) are mapped to the range (R). In other words, the function takes in three variables (x, y, and z) and outputs a single value in the real number system.

Now, for the function f(x,y,z) = xy, we can visualize its graph in the three-dimensional space by computing its level sets and sections.

Level sets are essentially the "slices" of the function at a constant output value. To compute the level sets, we set the function equal to a constant, say c, and solve for one of the variables. This will give us a two-dimensional curve in the three-dimensional space. For example, if we set xy = 1, we can solve for z to get z = 1/xy. This means that as long as xy = 1, the z-coordinate will be constant at 1/xy. Similarly, we can compute level sets for different values of c and plot them on the three-dimensional graph.

Sections, on the other hand, are the "cuts" of the function at a constant value of one of the variables. For example, if we set z = 0, we can see how the function behaves in the xy-plane, which is a two-dimensional section of the three-dimensional graph. We can also plot sections for different values of x or y to get a better understanding of the function's behavior.

In summary, level sets and sections help us visualize the graph of a function in a three-dimensional space by showing how it behaves at different constant values. I recommend practicing more problems to get a better understanding of these concepts. Good luck!