Answer to Question #138950 in Differential Geometry | Topology for shweta

Question

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

(a) f(x, y) = x2 + y2, c = 0, 1, 2, 3, 4.

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

Answer 1

a).Given a function f(x,y)

f(x,y) and a number c

c in the range of f,a

f,a level curve of a function of two variables for the value c

c is defined to be the set of points satisfying the equation f(x,y)=c.

1)c=0

(x²+y²)=0 =>(x,y)=(0,0)

2)c=1

(x²+y²)=1 -circle with center (0,0) and radius 1

3)c=2

(x²+y²)=2 -circle with center (0,0) and radius 2^1/2

4)c=3

(x²+y²)=3-circle with center (0,0) and radius 3^1/2

5)c=4

(x²+y²)=4 -circle with center (0,0) and radius 2

A graph is the family of circles, which equations I've described earlier.

A graph of the various level curves of a function is called a contour map.

b) 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 = -1 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.