contestada

Use Stokes' Theorem to evaluate S curl F · dS. F(x, y, z) = 4y cos z i + ex sin z j + xey k, S is the hemisphere x2 + y2 + z2 = 49, z ≥ 0, oriented upward.

Respuesta :

rodolforodriguezr

Answer:

-196π

Step-by-step explanation:

F can be rewritten as

[tex]\bf F(x,y,z)=(4ycos(z),e^xsin(z),xe^y)[/tex]

if S is the upper hemisphere  

[tex]\bf x^2 + y^2 + z^2 = 49[/tex]

oriented upward, then the border of S is the circle C

[tex]\bf x^2 + y^2 = 7^2[/tex]

traversed counterclockwise

By Stoke's theorem

[tex]\bf \displaystyle\iint_S(curl\;F)dS=\displaystyle\int_{C}F.dC [/tex]

where C is the circle of center (0,0,0) and radius 7 on the XY-plane  traveled counterclockwise.

This circle can be parametrized as

r(t) = (7cos(t), 7sin(t),0) with 0 ≤t ≤ 2π  

Computing the curve integral

[tex]\bf \displaystyle\int_{C}F.dC=\displaystyle\int_{0}^{2\pi}F(r(t))\bullet r'(t)dt=\displaystyle\int_{0}^{2\pi}F(7cos(t),7sin(t),0)\bullet (-7sin(t),7cos(t),0)dt=\\\\=\displaystyle\int_{0}^{2\pi} (28sin(t),0,7cos(t)e^{7sin(t)})\bullet(-7sin(t),7cos(t),0)dt=\\\\-196\displaystyle\int_{0}^{2\pi}sin^2(t)dt=\boxed{-196\pi}[/tex]

azikennamdi

The S Curl is computed at -196π. This was arrived at using the Stokes Theorem. The curl here is negative which means that the curl is clockwise.

What is the Stokes' Theorem?

The Theorem of Stokes indicates "The line integral of the particular vector function around a closed surface is identical to the surface integral of the curl of a function over that surface."

Step I

Let's recalibrate F as follows:

F (x,y,z) = (4yCos (z), eˣSin (z), x[tex]e^{y}[/tex])

Since S is the upper hemisphere we have our equation to be:

x²+y²+z² = 49

Where it is oriented in the vertical direction, then the frontier of S is the Circle C. Hence

x²+y² = 7².

See the attached for the full solution.

The curl here is negative which means that the curl is clockwise.

Learn more about the Stokes' Theorem at:

https://brainly.com/question/13105453

#SPJ5

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