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Surface integral of sphere

  1. Jan 13, 2015 #1
    1. The problem statement, all variables and given/known data
    find the values of the integral

    [itex] \int_{S} \vec A\cdot\ d\vec a [/itex]

    where,

    [itex] \vec A\ = (x^2+y^2+z^2)(x\hat e_{1}+y\hat e_{2}+z\hat e_{3}) [/itex]

    and the surface S is defined by the sphere [itex] R^2=x^2+y^2+z^2 [/itex]

    2. Relevant equations
    first i must evaluate the integral directly, so i don't think there are any specific formulas other than ones you must derive from the geometry specific to the problem. i also have to calculate using gauss' theorem but for that there's a simple equation.

    3. The attempt at a solution
    really looking for an explanation on surface integrals. i know that [itex] d\vec a [/itex] is a small area on the surface of the sphere and equations must be derived from the geometry. im having a hard time visualizing this and how it's suppose to work. for now, i would appreciate a good explanation of surface integrals to help me visualize the problem.

    thanks in advance.
     
  2. jcsd
  3. Jan 13, 2015 #2

    Orodruin

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    ##d\vec a## is the area element, which is an infinitesimal area multiplied by the unit normal of the surface. Try to answer the following: What is the unit normal of the sphere? What is the product of ##\vec A## with ##d\vec a##? How can you simplify the integrand using this?
     
  4. Jan 13, 2015 #3

    BvU

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    Hello Pizza, welcome to PF :)

    You familiar with spherical coordinates ? Reason I ask is because after all this is all happening on a sphere. Can you transform ##\vec A(x, y, z)## to ##\vec A(r,\theta, \phi)## ?

    [edit] Well, I see you got some good help already. Bedtime for me!
     
  5. Jan 13, 2015 #4
    [itex] d\vec a [/itex] is perpendicular to the surface of the area (2 possible directions).
     
  6. Jan 13, 2015 #5

    Orodruin

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    Yes, this is what I said, but which vector is perpendicular to the sphere surface at a given point on the sphere?
     
  7. Jan 13, 2015 #6
    I think Orodruin is asking "which spherical coordinate system unit vector is perpendicular to the sphere surface at a given point on the sphere?"

    What is your physical interpretation of the vector A at the surface of the sphere? In what direction is it pointing at each point over the surface?

    Chet
     
  8. Jan 13, 2015 #7
    [itex] \hat r [/itex] is the answer i believe you may be looking for. im not quite sure how to physically interpret [itex] \vec A [/itex]. i just know that it's some vector along the surface of the sphere. sorry if im making this difficult.
     
  9. Jan 13, 2015 #8

    Orodruin

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    Correct, so ##d\vec a = \hat r\, da## where da is the infinitesimal area. Now, can you write ##\vec A## in terms of the position vector ##\vec r##?
     
  10. Jan 13, 2015 #9
    [itex] \vec A = (x, y, z) \vec r [/itex] which i believe is equal to[itex] (x \hat e_{1} + y\hat e_{2} + z\hat e_{3}) [/itex] which is the second half of [itex] \vec A [/itex] mentioned in the problem
     
  11. Jan 13, 2015 #10
    What is your understanding of what (x2+y2+z2) represents at the surface of a sphere of radius r?

    Chet
     
  12. Jan 14, 2015 #11

    BvU

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    This [itex] \vec A = (x, y, z) \vec r [/itex] can't be right: what can possibly its components (if existent) ?

    Dear dude, I am starting to worry if you understand how ## (x \hat e_{1} + y\hat e_{2} + z\hat e_{3} )## is to be interpreted. Can you express it in Cartesian coordinates ? And in Spherical coordinates ?
     
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