# Use the Divergence Theorem to calculate the surface integral int int_S F · dS, that is, calculate the flux of F across S. F(x, y, z) = (x^3 + y^3)i + (y^3 + z^3)j + (z^3 + x^3)k, S is the sphere with center the origin and radius 2.

Use the Divergence Theorem to calculate the surface integral $\int {\int }_{S}F·dS$, that is, calculate the flux of F across S.
$F\left(x,y,z\right)=\left({x}^{3}+{y}^{3}\right)i+\left({y}^{3}+{z}^{3}\right)j+\left({z}^{3}+{x}^{3}\right)k$, S is the sphere with center the origin and radius 2.
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Bentley Leach
Let E be a simplesolid region and S is the boundary surface of E with positive orientation and Let $\stackrel{\to }{F}$ be a vector field whose components have continous first orderpartial derivatives. then
$\int {\int }_{S}\stackrel{\to }{F}\cdot d\stackrel{\to }{S}=\int {\int }_{E}\int ÷\stackrel{\to }{F}dV$
Step 2
Apply the above theorem as follows.
Given that $F\left(x,y,z\right)=\left({x}^{3}+{y}^{3}\right)\stackrel{\to }{i}+\left({y}^{3}+{z}^{3}\right)\stackrel{\to }{j}+\left({z}^{3}+{x}^{3}\right)\stackrel{\to }{k}$
div $\stackrel{\to }{F}=\mathrm{\nabla }\stackrel{\to }{F}=3{x}^{2}+3{y}^{2}+3{z}^{2}$
$\int {\int }_{S}\stackrel{\to }{F}\cdot d\stackrel{\to }{S}=\int {\int }_{E}\int idv\stackrel{\to }{F}dV$
$=\int {\int }_{E}\int \left(3{x}^{2}+3{y}^{2}+3{z}^{2}\right)dV$
$=3\int {\int }_{E}\int \left({x}^{2}+{y}^{2}+{z}^{2}\right)dV$
$=36\int {\int }_{E}\int \left(2\right)dV⟨{}^{\prime }{x}^{2}+{y}^{2}+{z}^{2}={2}^{2}\right)$
$=72\int {\int }_{E}\int 1dV$
$=72\left(\frac{4\pi }{3}×{2}^{3}\right)$ ($\because$ volume of the sphere is $\frac{4\pi }{3}{r}^{3}\right)$
$=768\pi$