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Vector calculus - Divergence Theorem

  1. Feb 16, 2007 #1
    1. The problem statement, all variables and given/known data

    Find [tex]\int_{s} \vec{A} \cdot d\vec{a}[/tex] given [tex]\vec{A} = ( x\hat{i} + y\hat{j} + z\hat{k} ) ( x^2 + y^2 + z^2 )[/tex] and the surface S is defined by the sphere [tex]R^2 = x^2 + y^2 + z^2[/tex] directly and by Gauss's theorem.

    2. Relevant equations

    [tex]\int_{s} \vec{A} \cdot d\vec{a} = \int_{V} \nabla \cdot \vec{A} da = \int\int\int \frac{1}{r^2} \frac{\partial}{\partial r}(r^2 A_{r}) r^2 \sin\theta dr d\theta d\phi = \int\int\int \frac{\partial}{\partial r}(r^2 A_{r}) \sin\theta dr d\theta d\phi = 4 \pi R^5[/tex]

    3. The attempt at a solution

    Given the answer, I was able to work backwards using limits of integration of 0 to 2pi for phi, 0 to pi for theta and 0 to R on r to find out that A = r^3. Now, the question I have is how can I show that A = r^3 given the A in cartesian coordinates? Since [tex]r=\sqrt{x^2 + y^2 + z^2}[/tex] its clear that [tex]A=r^2(x\hat{i} + y\hat{j} + z\hat{k})[/tex] but to call the vector valued term r is not agreeing with me. Obviously, since the surface is a sphere, I thought it would be easier to use spherical coordinates...

    Edit: Using cartesian coordinates I can find [tex]\nabla \cdot \vec{A} = 5(x^2 + y^2 + z^2) = 5r^2[/tex] but using the formula given for spherical coordinates, I would need to show A = r^3 to do a full solution in spherical.
    Last edited: Feb 17, 2007
  2. jcsd
  3. Feb 17, 2007 #2


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    What do you mean "A= r3"? What is A? There is no A in the statement of the problem.
  4. Feb 17, 2007 #3
    A is given, it is the vector. It's in the first line... Of course, to solve for A in spherical I worked backwards from the answer (I needed an R^5 in the final answer, and the divergence in spherical gives me an integral of a partial of r^2 * A, so obviously r^2 * A needs to have a 5th power, thus A = r^3, working it through, it does come out correctly).

    When I try to convert the given A into spherical directly, I can only get as far as [tex]\vec{A} = r^2(x\hat{i}+y\hat{j}+z\hat{k})[/tex]

    Another question I have is whether it would be easier to show it directly in cartesian or spherical...I'm leaning towards cartesian because A is given in cartesian.
    Last edited: Feb 17, 2007
  5. Feb 18, 2007 #4


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    As Halls pointed out, that's not right.

    The conversion from cartesian to spherical, and vice versa, should be found in any vector calculus text.

    (or try, http://www.math.montana.edu/frankw/ccp/multiworld/multipleIVP/spherical/body.htm)

    The integral [itex]\int_{s} \vec{A} \cdot d\vec{a}[/itex] over [itex]x^2 + y^2 + z^2=R^2[/itex] should be easier to evaluate in spherical coordinates because of the symmetry (ie, you're integrating over a sphere).
    Last edited: Feb 18, 2007
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