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Math Help - Spherical Coordinates Conundrum

  1. #1
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    Spherical Coordinates Conundrum

    Hello All,

    I'm working with surface integrals in spherical coordinates.

    The equation of the surface of the sphere is simply r, yes? (not to be confused with the equation of the surface area of a sphere 4.pi.r^2)

    When calculating the surface of a hemisphere, the equation of the curved surface of a hemisphere remains as r (not to be confused with the surface area of the hemisphere 2.pi.r^2)?

    What about the closed base of the hemisphere, how do I explain, where that is, in spherical terms? Is it in the manner that we explain a circle in cartesian coordinates, ie r = x^2 + y^2 (not to be confused with the surface area of a circle pi.r^2)

    I am really struggling with the understanding of the equation of the surface - I can carry out definite integrals quite happily for most shapes to find their volume/area, but I don't understand this surface equation?

    Hope someone can help.

    Kindest regards,
    Last edited by MaverickUK82; October 24th 2012 at 12:04 AM.
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  2. #2
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    Re: Spherical Coordinates Conundrum

    Hey MaverickUK82.

    There are a number of ways you can go about this: you can use the stuff in vector calculus or you can set up an area integral where you map the surface to what is called a chart and all this means is that you take the surface and stick it on a flat piece of paper.

    Mathematically this is just a transformation or a change of variables where your two variables are the north/south line and the east-west line where you are trying to find the area as if you took your surface and put it on a flat piece of paper like the following:

    http://www.classworksacademy.net/mir...projection.gif

    So you can either transform your surface in its local system (r,theta,phi) to something like the above and use regular integration (finding the area of the region) or you can use vector calculus and use what is called a surface integral. Both do the same thing and the surface integral actually does this projection for you by using the dot or inner products (which are just ways of finding specific kinds of projection components).

    Are you currently doing vector calculus?
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  3. #3
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    Re: Spherical Coordinates Conundrum

    Hey chiro, yep, It's vector calculus - I'm thinking this - if the hemisphere is of radius 1, the equation of the curved surface would be r=1 and the equation of the flat portion of the hemisphere, would be theta=1/2 pi , that's assuming the hemisphere is cut through the middle horizontally?
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  4. #4
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    Re: Spherical Coordinates Conundrum

    It's hard to say what you are getting at specifically: If you are looking at a function then the output is going to be (for cases like this) mapping R^n -> R^m. What kind of map are you talking about (please specify in terms of what the inputs are and what the outputs in both in vector form if you could)?
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