$Use Stokes's Theorem to evaluate $\iint_S ( ∇ × F ) ⋅ d S$ on the given surface$

Question

$Use Stokes's Theorem to evaluate $\iint_S ( ∇ × F ) ⋅ d S$ on the given surface$

Let $F = ( − x^ 3 , x y z , z ^3 )$, and S be the surface defined by $\{ ( x , y , z ) | x^2 + y^2 + z^2 = 3 , y \geq 0 \}$ . Evaluate $$\iint_S ( ∇ × F ) ⋅ d S$$,
using Stokes's Theorem.

Multivariable Calculus Stokes' Theorem Calculus

Ava Smith

31

Report

Answer

The answer is accepted.

Join Matchmaticians Affiliate Marketing Program to earn up to a 50% commission on every question that your affiliated users ask or answer.

Answer 1

Answers can only be viewed under the following conditions:

The questioner was satisfied with and accepted the answer, or
The answer was evaluated as being 100% correct by the judge.

View the answer

By Stokes's Theorem
\[ \iint_{S} (\nabla \times F)⋅dS=\int_{C}F \cdot dr,\]
where $C$ is a circle of radius $\sqrt{3}$ on $xz$-plane with positive orientation with respect to the surface $S$. Indeed we can parametrize $C$ by $C=(\sqrt{3}\cos \theta, 0, \sqrt{3} \sin \theta)$, $-2\pi \leq \theta \leq 0$. Hence

\[\int_{C}F \cdot dr=\int_C (-x)^3dx+z^3dz=\int_{-2\pi}^{0}-(\sqrt{3})^3 \cos ^3 \frac{dx}{d \theta}+(\sqrt{3})^3 \sin^3 \theta \frac{dz}{d \theta} d \theta\]
\[=(\sqrt{3})^4\int_{-2\pi}^{0}\cos ^3 \sin \theta +\sin ^3 \theta \cos \theta d \theta\]
\[=9 \int_{-2\pi}^{0} \sin \theta \cos \theta (\sin^2 \theta +\cos ^2 \theta) d \theta\]
\[=9 \int_{-2\pi}^{0} \sin \theta \cos \theta d \theta=\frac{9}{2} \int_{-2\pi}^{0} \sin 2\theta d \theta\]
\[\frac{9}{4} (-\cos 2 \theta)|_{-2 \pi}^{0}=\frac{9}{4} (-1+1)=0.\]

Erdos

4.8K

$Use Stokes's Theorem to evaluate $\iint_S ( ∇ × F ) ⋅ d S$ on the given surface$

Answer

Related Questions

Search