Mathematics: Use Stokes Theorem to evaluate S curl F · dS. F(x, y, z) = x2y3zi + sin(xyz)j + xyzk, S is the part of the cone y2 = x2 + z2 that lies between the planes y = 0 and y = 3, oriented in the direction of the positive y-axis.

Use Stokes Theorem to evaluate S curl F · dS. F(x,

Ответ:

Jadas960

31.01.2022

Mathematics

$\frac{2187 \pi}{4}$

Step-by-step explanation:

Using Stoke's theorem, we get

$\int \int_S curl F.dS = \int_C F.dr$

Now parametrize C

r(t) = ( 3 cos t, 3 - 3 sin t ) where, t in [0 2π]

$\int_C F. dr= \int_0 ^{2 \pi}F ( 3 \cos t, 3 - 3 \sin t). r'(t)dt$

$= \int_0^{2 \pi}(( 9 \cos^2 t)(27)(-3\sin t), \sin t \times (-27 \cos t \sin t)) . (-3 \sin t, 0 - 3 \cos t) dt$

$= 27 \int_0^{2 \pi} 81 \cos ^2 t \sin ^2 t + 3 \cos ^2 t \sin tdt$

$= 27 \int_0^{2 \pi} \frac{81}{4} \sin ^2 2t+27 [ - \cos^3 t]_0^{2 \pi}$

$= 27 \times \frac{81}{4} \int_0^{2 \pi} \frac{(1- \cos 4t)}{2}$

$= \frac{2187}{8}[t-\frac{1}{4} \sin 4t]_0^{2 \pi}$

$= \frac{2187 \pi}{4}$

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Ответ:

katherineweightman

21.01.2020

Mathematics

Let's call the distance between the moon and sun as z. The scientist can calculate the numerical value of sin x° and then write sin x° as y / z. They can then set the numerical value of sin x° equal to y / z and then solve for z.

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Use Stokes' Theorem to evaluate S curl F · dS. F(x, y, z) = x2y3zi + sin(xyz)j + xyzk, S is the part of the cone y2 = x2 + z2 that lies between the planes y = 0 and y = 3, oriented in the direction of the positive y-axis.

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