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Math Help - what is the mistake here

  1. #1
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    what is the mistake here

    why does the prof wrote a mistake there and wrote 2cos theta in there
    ??

    (i didnt typed the question correctly,the original is
    i need to calculate the integral on the volume enclosed by z>=0 and the sphere wich is written in the photo.)

    ??
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  2. #2
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    Re: what is the mistake here

    Your prof is correct; you need to substitute spherical coordinates into the equation for the volume. Your limits are wrong because you are not integrating over a sphere of radius 2 centered at the origin.
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  3. #3
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    Re: what is the mistake here

    correct i am integration on sphere centered (0,0,1)

    in spherecal coordinated the third coordinate is the radius and from the photo and you said it youself that the radius is 2.

    when i change the coordinated from cartesian to spherecal
    i do it from the sketch,so i get radius from 0 till 2
    cant see how the shifted center changes the intervals the way it did.
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  4. #4
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    Re: what is the mistake here

    The spherical coordinates are with respect to the Cartesian axes, not the center of the sphere that you're integrating over. Substituting the expressions for the spherical coordinates into x^2+y^2+z^2\le 2z gives \rho\le 2\cos \phi.

    You can shift the Cartesian coordinates to the center of the sphere but then you need to make the corresponding change to the integrand.
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  5. #5
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    Re: what is the mistake here

    Quote Originally Posted by ojones View Post
    The spherical coordinates are with respect to the Cartesian axes, not the center of the sphere that you're integrating over. Substituting the expressions for the spherical coordinates into x^2+y^2+z^2\le 2z gives \rho\le 2\cos \phi.

    You can shift the Cartesian coordinates to the center of the sphere but then you need to make the corresponding change to the integrand.
    ok ,but how mathematicky can i get this transition
    how to get this expression
    \rho\le 2\cos \phi
    Last edited by transgalactic; July 1st 2011 at 03:40 AM.
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  6. #6
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    Re: what is the mistake here

    You did something similar already on your exam; how else did you transform the integrand to 1/\rho?
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  7. #7
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    Re: what is the mistake here

    Quote Originally Posted by ojones View Post
    You did something similar already on your exam; how else did you transform the integrand to 1/\rho?
    why would i want
    1/\rho
    ?
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  8. #8
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    Re: what is the mistake here

    My point is how did you get 1/\rho?
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  9. #9
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    Re: what is the mistake here

    Quote Originally Posted by ojones View Post
    My point is how did you get 1/\rho?
    \rho^2=x^2+y^2+z^2
    so i got it from putting the \rho instead of the expression in the denominator
    cant see how it changes the interval into this cosine on top
    ?
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  10. #10
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    Re: what is the mistake here

    You skipped a big step here. How is this identity obtained?
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  11. #11
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    Re: what is the mistake here

    i have some linking formulas
    between spherecal and cartesian
    but we dont use them
    we just see the sketch and build the integral by it.

    can you tell me from the transision formulas
    why \rho
    have these intervals?
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  12. #12
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    Re: what is the mistake here

    You don't use the expressions relating spherical to cartesian coordinates?? That's absurd. This is why you don't understand your professor's comments. Go back to the beginning of this subject and learn it properly this time. There are no shortcuts, believe me.
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  13. #13
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    Re: what is the mistake here

    i have some linking formulas
    between spherecal and cartesian
    but we dont use them
    we just see the sketch and build the integral by it.

    can you tell me from the transision formulas
    why \pho
    have these intervals?

    you dont use these formulas to build the integral
    you do it by the sketch

    but here i dont know how to get the 2cosine interval
    could you show me the transition?
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  14. #14
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    Re: what is the mistake here

    I'll give you a hint. What is z in spherical coordinates?
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  15. #15
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    Re: what is the mistake here

    z=\rho cos \theta
    what to do now?
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