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How to inverse surface integral of a vector field

  1. Aug 17, 2010 #1
    Assume that I know the value of [tex]\iint_{S} \overrightarrow{F} \cdot \hat{n} dS[/tex] over any surface in [tex]\mathbb{R}^3[/tex], where [tex]\overrightarrow{F}(x,y,z)[/tex] is a vector field in [tex]\mathbb{R}^3[/tex] and [tex]\hat{n}[/tex] is the normal to the surface at any point considered.
    Using that I would like to compute [tex]\overrightarrow{F}[/tex]. How can this be done?

    P.S. I think I suffices to know [tex]\iint_{S} \overrightarrow{F} \cdot \hat{n} dS[/tex] over the planes perpendicular to the axes.
     
    Last edited: Aug 17, 2010
  2. jcsd
  3. Aug 17, 2010 #2
    Consider a small circumference centered at a point x_0, with radius R and normal directed along a generic direction n (|n|=1):

    [tex]S(\mathbf^{x}_0,R,\mathbf{n})=\{\mathbf{x}\in\mathbb{R}^3:(\mathbf{x}-\mathbf{x}_0)\cdot\mathbf{n}=0,|\mathbf{x}-\mathbf{x}_0|<R\}[/tex]

    then, if F is sufficiently smooth, you can calculate it's components

    [tex]\mathbf{F}(\mathbf{x}_0)\cdot\mathbf{n}=\lim_{R\rightarrow 0}\,\,\frac{1}{\pi R^2}\,\,\int_{S(\mathbf^{x}_0,R,\mathbf{n})} \mathbf{F} \cdot d\mathbf{S}[/tex]
     
    Last edited: Aug 17, 2010
  4. Aug 17, 2010 #3
    Great!
    This works perfectly. Thank You.
     
  5. Aug 18, 2010 #4
    Glad to help. Note that you are not forced to take small circumferences, but every small surface you like, depending on what's more useful: squares, triangles...
     
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