I am learning about level sets and it was introduced by saying that a function with 3 variables f(x, y, z) will have a graph in R4 (4th dimension), which makes it necessary to have a way of visualizing it (using level sets).

Then what is the purpose of drawing the level sets for a function with 2 variables f(x, y), where the graph will be in R3 (3rd dimension) which can easily be visualized/drawn?

  • $\begingroup$ Have you ever seen a topographic (contour) map? $\endgroup$ – amd Oct 19 '17 at 4:05
  • $\begingroup$ yes but I guess I understand the purpose of that more ? $\endgroup$ – mathguy Oct 19 '17 at 4:05

If you genuinely can visualize or draw a three-dimensional function accurately, then you're a better mathematician than I - I can handle $z = x^2 + y^2$ or something similar, but (for example) $z = x^3 - 6xy + 2x - y^2 + 4$? By constructing level sets, we can assemble a picture of something we can't visualize well. If I needed to, I could make level sets for $z = x^3 - 6xy + 2x - y^2 + 4$ by hand, and get a pretty good mental image of the surface.

Also, additional data can be acquired from level sets. In particular, the direction of the gradient vector is often difficult to spot by looking at a three-dimensional surface, but is easy to see by looking at a level set.

One last thing: Even if you can visualize a 3-D surface accurately, you can't get the whole picture all at once - this is the same reason we still use maps, even though the average phone is perfectly capable of displaying a 3-D image of the area. The point of a map is to see the whole region, without obstacles like buildings and mountains getting in the way. If you look at a 3-D surface, you'll often see that one part of it obscures other parts, and you have to keep rotating the image to get a sense of the whole thing. Using level sets, you can get a sense of the whole surface at once.

  • $\begingroup$ thanks, i was starting to understand and you explained well $\endgroup$ – mathguy Oct 19 '17 at 4:16

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