Mapping the Unit Ball to a Sphere of Radius c

[Jadehaan]

Homework Statement

Let B be the outside of the unit ball centered at the origin, and let c be a non-zero constant. Consider the mapping

where k=1,2,3.
Find the image of the set B under the mapping. (Hint: consider the norm of (y₁, y₂, y₃))

Homework Equations

The unit ball would be 2 dimensional so the formula would be x²+y²=1

The Attempt at a Solution

I have no attempt at the solution. I apologize for the lack of effort, I am confused at where to start, I have never encountered a problem formatted like this.

Thanks for any tips you can give me,
Jim

 

[lanedance]

start by considering one variable say y1

 

[lanedance]

clicked a little early...

so start with one variable, say y₁, consider 2 points on the x₁ axis, the first with x₁ on the surface of the ball, and the second as x₁ tends to infinity, see where they are mapped to and have think about the points in between

as hinted what can you say about the norm of y₁... is it bounded? and how does it behave as you move along say the x₁ axis

 

[Jadehaan]

$$y_{1}=\frac{cx_1}{x_1^2+x_2^2+x_3^2}$$
If x is on the unit ball, x=$$\sqrt{1-y^2}$$
How do $$x_1, x_2, x_3$$ play a part?

 

[Dick]

Why do you think the ball is two dimensional? Looks to me like it should be three dimensional with (x1,x2,x3) being the coordinates.

 

[Jadehaan]

Right, I misdefined the unit ball in my last two posts. Now the unit ball should be defined by $$x^{2}_{1}+x^{2}_{2}+x^{2}_{3}=1$$

Using the result for $$y_{1}$$ if $$x_{1}$$ is on the surface of the ball, $$x_{1}=1$$ and the other x's must equal 0. This implies that $$y_{1}=c$$. And if $$x_{1}$$ grows large, the other x's grow small, until $$y_{1}=\frac{c}{\infty}=0$$

In this case, this would apply for every x, which let's me think that I'm not doing it right. Do I consider the unit ball as $$y^{2}_{1}+y^{2}_{2}+y^{2}_{3}=1$$? Is the norm of $$|y_{1}|=\frac{c|x_{1}|}{|x_{1}|^2+x^{2}_{2}+x^{2}_{3}}$$? How would the norm affect the mapping?

 

[Dick]

Think about what happens to spheres, if |(x1,x2,x3)|=sqrt(x1^2+x2^2+x3^2)=1, i.e. (x1,x2,x3) is on a sphere of radius 1, then doesn't (y1,y2,y3) lie on a sphere of radius c? What happens if (x1,x2,x3) lies of a sphere of radius 2? Doesn't (y1,y2,y3) lie on a sphere of radius c/2?