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Let V be the solid sphere of radius centered at the origin. Let S be the surface of V oriented with outward unit normal. Consider the vector field

(ri + yj + zk) F(T,y,2) = (2x^2 + y^2 + 2z^2)^(3/2)

Evaluate the flux integral

âˆ«âˆ«âˆ« F Â· n dS

by direct calculation.

Evaluate

âˆ«âˆ«âˆ« V Â· F dV

by direct calculation.

(c) Compare your answers to parts (a) and (b) and explain why Gauss’ theorem does not apply.

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00:10

Use the Divergence Theorem to calculate the surface integral $ \iint_S \textbf{F} \cdot d\textbf{S} $; that is, calculate the flux of $ \textbf{F} $ across $ S $.

$ \textbf{F} = | \textbf{r} |^2 \textbf{r} $, where $ \textbf{r} = x \, \textbf{i} + y \, \textbf{j} + z \, \textbf{k} $,$ S $ is the sphere with radius $ R $ and center the origin

02:10

1.7.3.a. Find the divergence of the vector field F(x, y, z) = (e^x + y^2)i + Tye^zj + (2 – 5)k.b. Use the divergence theorem and the change of variables formula for a multiple integral to evaluate the flux integral âˆ® F Â· dA, where S is the boundary of the region D = {(x, y, z) âˆˆ R^3 | (2x + 2)^2 + (y + 2)^2 + 2^2 < 1} and N is the outward-pointing unit normal. (You may use the fact that the volume bounded by the unit sphere is (4/3)Ï€).

00:08

$ \textbf{F}(x, y, z) = (x^3 + y^3) \, \textbf{i} + (y^3 + z^3) \, \textbf{j} + (z^3 + x^3) \, \textbf{k} $, $ S $ is the sphere with center the origin and radius 2

03:49

Use the Divergence Theorem to calculate the surface integral âˆ¬F Â· dS; that is, calculate the flux of F across S.F(x, y, z) = (xÂ³ + yÂ³)i + (yÂ³ + zÂ³)j + (zÂ³ + xÂ³)k,S is the sphere with center at the origin and radius 2.

02:16

Use the Divergence Theorem to calculate the surface integral âˆ¬S F Â· dS; that is, calculate the flux of F across S. F = (x^2 + y^2 + z^2)^(1/2); where r = xi + yj + zk. S is the sphere with radius R and center at the origin.

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