Answer to Question #138975 in Differential Geometry | Topology for shweta

Question

Sketch the level curves f−1(c) for the following functions:

f(x, y, z) = x − y2 − z2, c = −1, 0, 1.

Answer 1

f(x, y, z) = x − y² − z², c = −1, 0.

1)c=-1

x − y² − z²=-1

x+1=y² +z²- elliptical paraboloid, formed by rotation around the x axis

2)c=0

x − y² − z² = 0

x= y² +z²- elliptical paraboloid, formed by rotation around the x = axis

A graph is the family of elliptical paraboloids, where the top is shifts along the x-axis, depends on c-value.

3)c=1

x-1=y² +z²- elliptical paraboloid, formed by rotation around the x axis