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Math Help - Flux Field Across the Boundary of the Sphere

  1. #1
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    Flux Field Across the Boundary of the Sphere

    i'm kinda stuck with this question and dont know where to begin solving.

    Work out the outward flux of the field  \vec{F}=xz\hat{i}-zy\hat{j}+yz\hat{k} across the boundary of the sphere x^2+y^2+z^2=1. Verify your result using Divergence Theorem.

    Since it's the vector calculus, i posted this thread here. hope, didn't made it wrong.
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  2. #2
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    Quote Originally Posted by Lafexlos View Post
    i'm kinda stuck with this question and dont know where to begin solving.

    Work out the outward flux of the field  \vec{F}=xz\hat{i}-zy\hat{j}+yz\hat{k} across the boundary of the sphere x^2+y^2+z^2=1. Verify your result using Divergence Theorem.

    Since it's the vector calculus, i posted this thread here. hope, didn't made it wrong.
    It would have been better to post what you have tried so we could see where you were stuck. There are many ways of doing a problem like this and we cannot be sure which is appropriate for you. In any case, here is how I would do this problem.

    Using spherical coordinates, with \rho equal to the radius of the sphere, 1, we can write the "position vector" of any point on the surface as
    \vec{r}(\theta, \phi)= cos(\theta)sin(\phi)\vec{i}+ sin(\theta)sin(\phi)\vec{j}+ cos(\phi)\vec{k}

    Differentiating with respect to \theta and \phi gives two vectors in the tangent plane.
    \vec{r}_\theta= -sin(\theta)sin(\phi)\vec{i}+ cos(\theta)sin(\phi)\vec{j}
    \vec{r}_\phi= cos(\theta)cos(\phi)\vec{i}+ sin(\theta)cos(\phi)\vec{k}- sin(\phi)\vec{k}

    The cross product of those two tangent vectors, the "fundamental vector product" for this surface, gives the "vector differential of surface area",
    d\vec{S}= \left(cos(\theta)sin^2(\phi)\vec{i}+ sin(\theta)sin^2(\phi)\vec{j}+ sin(\phi)cos(\phi)\right)d\theta d\phi
    (The order in which you take the cross product determines the sign. I chose the way that gives an outward pointing normal [all components positive] because the problem said "the outward flux".)

    You want to integrate \int\int \vec{F}\cdot d\vec{S} over the surface of the sphere: \theta from 0 to 2\pi, \phi from 0 to \pi. Of course, you will need to change \vec{F} to these coordinates.
    Last edited by HallsofIvy; April 5th 2011 at 09:28 AM.
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  3. #3
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    i was stuck at finding d\vec{S} and now i understand the way and see why i couldn't solve.
    i put \phi at wrong place hence i got \vec{r}(\theta, \phi)= cos(\theta)cos(\phi)\vec{i}+ sin(\theta)cos(\phi)\vec{j}+ sin(\phi)\vec{k} as position vector.
    Thanks for help.
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