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Vector Calculus Question

  1. Apr 9, 2005 #1
    Anyone take a look at this vector calculus question for me:

    Q. If n is the uni normal to the surface S, evaluate Double Integral r.n dS over the surface of a sphere of radius 'a' centred at the origin.

    So I did:

    r = (x,y,z)
    Sphere: x^2 + y^2 + z^2 = a^2

    let f = x^2 + y^2 + z^2

    n = gradf / |grad f|

    therefore n = (x,y,z)/a

    n.r = a

    Now how do I proceed with the integral? I thought it would just be

    int(2pi->0) int (pi->0) int(a->0) a r.dr.d[theta].d[phi]

    which gives the answer [pi]^2.[a]^3 which really doesn't look right! I think it's the actual integral I've made a mistake with! HELP!!!! I HATE VECTOR CALCULUS [and I really need to learn to use latex!]
  2. jcsd
  3. Apr 9, 2005 #2
    Isnt n = [tex] \frac{2x,2y,2z}{|gradF|} [/tex]?
  4. Apr 9, 2005 #3
    Btw, I know that a simple way to solve my problem is just to use the fact that the surface area of the sphere is 4*pi*a^3, but I want to solve the integral explicitly (as practise). Have found my tutor's notes and she found the 'area element' to be a*sin^2*theta*d[theta]*d[phi] and thus carried out a double integral. Could someone explain this please?
  5. Apr 9, 2005 #4

    Yes but the |gradF| on the bottom gives you a factor of 2, so it cancels :frown: Having found my tutor's notes I know that I'm doing fine up until the integral and I shouldn't be doing it as a triple integral, but rather a double integral as I've explained in my above post.
  6. Apr 9, 2005 #5
  7. Apr 9, 2005 #6
    thanks. they're still carrying out triple integrals though, whereas I should only be doing a double integral :s
  8. Apr 9, 2005 #7
    I'd help you out but I dont have my calc 3 book to recall the specific steps in deriving spherical coordinates, but im pretty sure spherical coordinates only works in triple integrals since it uses 3 parameters, p,theta,phi.

    Im probably not the person to helping you with this. Sorry.
  9. Apr 9, 2005 #8
    Last edited: Apr 9, 2005
  10. Apr 9, 2005 #9


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    Problem requires evaluation of {∫ ∫ r⋅dA} over Surface of Sphere of Radius "a":
    The Unit Area normal element on the Sphere's surface is given by:
    dA = r2sin(φ)⋅dθ⋅dφ⋅r/|r|
    ::: ⇒ ∫ ∫ r⋅dA = ∫ ∫ r⋅r2sin(φ)⋅dθ⋅dφ⋅r/|r| =
    = ∫ ∫ r3sin(φ)⋅dθ⋅dφ = ???
    The above Double Integral should be evaluated at constant (r = a) for integration limits {0 ≤ θ ≤ 2*π} and {0 ≤ φ ≤ π}.

    For more info, see Equation #14 at:

    Last edited: Apr 9, 2005
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