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

Question #138950

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.


1
Expert's answer
2020-10-20T18:29:49-0400

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 21/2

4)c=3

(x2+y2)=3-circle with center (0,0) and radius 31/2

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 − z= 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.


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