Answer to Question #138615 in Differential Geometry | Topology for anjali g

Question #138615
Sketch the level curves f−1(c) for the following functions:
a) f(x ,y ,z) = x−y^2 −z^2, c = −1,0,1.
1
Expert's answer
2020-10-18T17:55:18-0400

Given,

"f(x,y,z)=x-y^2-z^2"

Now, level curve is defined as

"G_f=\\{(x,y,z)\\in \\mathbb{R}^3|f(x,y,z)=c\\}"

That is, here, "f^{-1}(c)=G_f"

Now, set

"f(x,y,z)=c\\implies (x-c)=y^2+z^2"

Clearly, the level curves are family of paraboloid which passes through the center "(c,0,0)"

Here, blue, green and red are level curve of "f^{-1}(-1),f^{-1}(0)\\& f^{-1}(1)" respectively.


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