Question

Let $\mathbf{n}$ be the outer unit normal (normal away from the origin) of the parabolic shell $$ \text { S: } 4 x^{2}+y+z^{2}=4, \quad y \geq 0 $$ and let $$ \mathbf{F}=\left(-z+\frac{1}{2+x}\right) \mathbf{i}+\left(\tan ^{-1} y\right) \mathbf{j}+\left(x+\frac{1}{4+z}\right) \mathbf{k} $$ Find the value of $$ \iint_{S} \nabla \times \mathbf{F} \cdot \mathbf{n} d \sigma $$

          Let $\mathbf{n}$ be the outer unit normal (normal away from the origin) of the parabolic shell
$$
\text { S: } 4 x^{2}+y+z^{2}=4, \quad y \geq 0
$$
and let
$$
\mathbf{F}=\left(-z+\frac{1}{2+x}\right) \mathbf{i}+\left(\tan ^{-1} y\right) \mathbf{j}+\left(x+\frac{1}{4+z}\right) \mathbf{k}
$$
Find the value of
$$
\iint_{S} \nabla \times \mathbf{F} \cdot \mathbf{n} d \sigma
$$

Added by Shane T.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Adi S

09/15/2023

Step 1

$$\nabla \times \mathbf{F} = \left(\frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z}\right) \times \left(-z+\frac{1}{2+x}, \tan^{-1}y, x+\frac{1}{4+z}\right)$$ $$\nabla \times \mathbf{F} = \left(\frac{\partial}{\partial Show more…

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Let $\mathbf{n}$ be the outer unit normal (normal away from the origin) of the parabolic shell $$ \text { S: } 4 x^{2}+y+z^{2}=4, \quad y \geq 0 $$ and let $$ \mathbf{F}=\left(-z+\frac{1}{2+x}\right) \mathbf{i}+\left(\tan ^{-1} y\right) \mathbf{j}+\left(x+\frac{1}{4+z}\right) \mathbf{k} $$ Find the value of $$ \iint_{S} \nabla \times \mathbf{F} \cdot \mathbf{n} d \sigma $$