Question

Use the Divergence Theorem to calculate the surface integral $$ \iint_S \mathbf{F} \cdot d\mathbf{S} $$ that is, calculate the flux of F across S. $$ \mathbf{F}(x, y, z) = 3xy^2\mathbf{i} + xe^z\mathbf{j} + z^3\mathbf{k} $$ S is the surface of the solid bounded by the cylinder $$ y^2 + z^2 = 4 $$ and the planes $$ x = -2 $$ and $$ x = 2 $$ Step 1 If the surface S has positive orientation and bounds the simple solid E, then the Divergence Theorem tells us that $$ \iint_S \mathbf{F} \cdot d\mathbf{S} = \iiint_E \text{div } \mathbf{F} \, dV $$ For $$ \mathbf{F}(x, y, z) = 3xy^2\mathbf{i} + xe^z\mathbf{j} + z^3\mathbf{k} $$ we have div F =

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

$$ \mathbf{F}(x, y, z) = 3xy^2\mathbf{i} + xe^z\mathbf{j} + z^3\mathbf{k} $$
S is the surface of the solid bounded by the cylinder $$ y^2 + z^2 = 4 $$ and the planes $$ x = -2 $$ and $$ x = 2 $$

Step 1

If the surface S has positive orientation and bounds the simple solid E, then the Divergence Theorem tells us that

$$ \iint_S \mathbf{F} \cdot d\mathbf{S} = \iiint_E \text{div } \mathbf{F} \, dV $$

For $$ \mathbf{F}(x, y, z) = 3xy^2\mathbf{i} + xe^z\mathbf{j} + z^3\mathbf{k} $$ we have

div F =

use the divergence theorem to calculate the surface integral iints mathbff cdot dmathbfs that is calculate the flux of f across s mathbffx y z 3xy2mathbfi xezmathbfj z3mathbfk s is the surf 07364

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