# Purpose of Level Curves

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?

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