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Tripple integrals, converting into spherical coordinates issue~

  1. Mar 5, 2007 #1
    Hello everyone, this is an example out of the book, but i'm confused on how they got the spheircal cordinates.

    Here is the problem:
    Evaluate tripple integral over B (x^2+y^2+z^2) dV and use spherical coordinates.

    Well the answer is the following:
    In spherical coordinates B is represented by {(p,theta,phi)| 0 <= p <= 1, 0 <= theta <= 2pi; 0 <= phi <= pi }; Thus ripple integral of B (x^2+y^2+z^2) dV = tripple intgral (p)^2*p^2 sin(phi) dp d(theta) d(phi)

    I'm lost on how they got (p)^2*p^2 sin(phi)

    I know the following though,
    [​IMG] [Broken]


    I figured out how they get the new bounds in spherical coordinates.

    But when I used the formula, all i got was
    (psin(phi)*cos(theta))^2 + (psin(phi)*sin(theta))^2 + ((pcos(phi))^2; not what they got.

    I also saw that: [​IMG] [Broken]
    but this still doesn't explain the extra p^2*sin(phi) it does explain the extra p^2 though.
    Any help would be great!
     
    Last edited by a moderator: Apr 22, 2017 at 4:10 PM
  2. jcsd
  3. Mar 5, 2007 #2
    The simple explanation is that a small change in x y and z is not exactly the same change in rho psi and theta. There must be some factor to correct for this.

    This explains it nicely:

    http://www.maths.abdn.ac.uk/~igc/tch/ma2001/notes/node77.html [Broken]
     
    Last edited by a moderator: Apr 22, 2017 at 4:10 PM
  4. Mar 5, 2007 #3
    I was able to follow the directions on converting to polar coordinates which I have no problem in doing but when I read the small part on spherical I really didn't see them show any conversion at all, i'll try to google to find more tutorials.
     
  5. Mar 6, 2007 #4

    HallsofIvy

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    In general, if you convert from x,y,z coordinates to u(x,y,z), v(x,y,z), w(x,y,z), then dxdydz= J(u,v,w)dudydz where J(u,v,w) is the "Jacobian" determinant:
    [tex]\left|\begin{array}{ccc}\frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} & \frac{\partial x}{\partial w} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} & \frac{\partial y}{\partial w} \\ \frac{\partial z}{\partial u} & \frac{\partial z}{\partial v} & \frac{\partial z}{\partial w}\end{array}\right|[/tex]

    In particular, for spherical coordinates, that is
    [tex]\left|\begin{array}{ccc}cos(\theta)sin(\phi) & -\rho sin(\theta)sin(\phi) & \rho cos(\theta)cos(\phi) \\ sin(\theta)sin(\phi) & \rho cos(\theta)sin(\phi) & \rho sin(\theta) cos(\phi) \\ cos(\phi) & 0 & -\rho sin(\phi)\end{array}\right|= \rho^2 sin(\phi)[/tex]

    I'm sure that's covered in your text book.
     
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