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Mapping & Change of Variables

  1. Nov 28, 2009 #1
    1. The problem statement, all variables and given/known data
    Let B be the outside of the unit ball centered at the origin, and let c be a non-zero constant. Consider the mapping latex2png.2.png where k=1,2,3.
    Find the image of the set B under the mapping. (Hint: consider the norm of (y1, y2, y3))

    2. Relevant equations

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

    3. 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
     
  2. jcsd
  3. Nov 28, 2009 #2

    lanedance

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    start by considering one variable say y1
     
  4. Nov 28, 2009 #3

    lanedance

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    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
     
    Last edited: Nov 29, 2009
  5. Nov 29, 2009 #4
    [tex]y_{1}=\frac{cx_1}{x_1^2+x_2^2+x_3^2}[/tex]
    If x is on the unit ball, x=[tex]\sqrt{1-y^2}[/tex]
    How do [tex]x_1, x_2, x_3[/tex] play a part?
     
  6. Nov 29, 2009 #5

    Dick

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    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.
     
  7. Nov 29, 2009 #6
    Right, I misdefined the unit ball in my last two posts. Now the unit ball should be defined by [tex]x^{2}_{1}+x^{2}_{2}+x^{2}_{3}=1[/tex]

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

    In this case, this would apply for every x, which lets me think that I'm not doing it right. Do I consider the unit ball as [tex]y^{2}_{1}+y^{2}_{2}+y^{2}_{3}=1[/tex]? Is the norm of [tex]|y_{1}|=\frac{c|x_{1}|}{|x_{1}|^2+x^{2}_{2}+x^{2}_{3}}[/tex]? How would the norm affect the mapping?
     
  8. Nov 30, 2009 #7

    Dick

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    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?
     
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