Question
Asked Dec 9, 2019
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Use the divergence theorem to find the outward flux || (F-n)dS, where the vector field
is given by F= /x² + y² + z² (xi+ yj+zk) and S is the surface of a closed region D
bounded by the concentric spheres x' + y' + z² = 4 and x' + y° +z? = 1.
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Use the divergence theorem to find the outward flux || (F-n)dS, where the vector field is given by F= /x² + y² + z² (xi+ yj+zk) and S is the surface of a closed region D bounded by the concentric spheres x' + y' + z² = 4 and x' + y° +z? = 1.

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Step 1

Given that the vector field,

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F (x, y,z)=(x + y² + z° )(xi + yj + zk) and S is the surface of a closed region D bounded by concentric spheres, x² + y? + z? = 4 and x² +y* +z° =1

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Step 2

Using the formula of outward flux of vector field,

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J[F-n ds Using the Gauss Divergence theorem, |[F-n ds = [[[ div FdV (Vx² + y² + z² )(xi + yj + zk) div F = V.F =|i- + j- +k Ôx *Vx* + y? +z° +- + y? +z? + y? +z? y. ây 4(x² + y² + z² ) Vx? + y? + z? 4x² + y² + z? dV = dxdydz

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Step 3

Then, the integra...

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SI[ div FdV = 4f[[ \x* + y² +z°dxcydz Now using the spherical coordinates, x = r cos esin Ø y = r sin O sin ø z = rcos Ø dxdydz = r* sin ødrd®dø Then, the integral becomes 27 4|||r* sin ødrdedø = 4|2 sin ødrdedø 0=0 Je=0 Jr=

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