Question

Compute the flux \iint_\Sigma \mathbf{F} \cdot d\mathbf{S} for the given vector field and surface. Confirm that a relevant theorem applies before using it. Compute directly if time. 2. $\mathbf{F}(x, y, z) = (x + y^2 z^2)\mathbf{i} + (y + z^2 x^2)\mathbf{j} + (z + x^2 y^2)\mathbf{k}$ where $\Sigma$ is the disk $x^2 + y^2 \le 4$ at $z = 4$ and the paraboloid $x^2 + y^2 = z$ for $z \le 4$, oriented outward. Multivariable Calculus (APMA E2000) 3. $\mathbf{F}(x, y, z) = x^3 \mathbf{i} + y^2 \mathbf{j} + 3z \mathbf{k}$ $\Sigma$ is the portion of the cylinder $x^2 + y^2 = 1$ bounded by the planes $z = 0$ and $z = 1.$

          Compute the flux
\iint_\Sigma \mathbf{F} \cdot d\mathbf{S}
for the given vector field and surface. Confirm that a relevant theorem applies before using
it. Compute directly if time.
2.
$\mathbf{F}(x, y, z) = (x + y^2 z^2)\mathbf{i} + (y + z^2 x^2)\mathbf{j} + (z + x^2 y^2)\mathbf{k}$
where $\Sigma$ is the disk $x^2 + y^2 \le 4$ at $z = 4$ and the paraboloid $x^2 + y^2 = z$ for $z \le 4$,
oriented outward.
Multivariable Calculus (APMA E2000)
3.
$\mathbf{F}(x, y, z) = x^3 \mathbf{i} + y^2 \mathbf{j} + 3z \mathbf{k}$
$\Sigma$ is the portion of the cylinder $x^2 + y^2 = 1$ bounded by the planes $z = 0$ and $z = 1.$

Compute the flux
∬Σ𝐅 ·d𝐒
for the given vector field and surface. Confirm that a relevant theorem applies before using
it. Compute directly if time.
2.
𝐅(x, y, z) = (x + y^2 z^2)𝐢 + (y + z^2 x^2)𝐣 + (z + x^2 y^2)𝐤
where Σ is the disk x^2 + y^2 ≤ 4 at z = 4 and the paraboloid x^2 + y^2 = z for z ≤ 4,
oriented outward.
Multivariable Calculus (APMA E2000)
3.
𝐅(x, y, z) = x^3 𝐢 + y^2 𝐣 + 3z 𝐤
Σ is the portion of the cylinder x^2 + y^2 = 1 bounded by the planes z = 0 and z = 1.

Added by Gema B.

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