noathequeen

noathequeen

04.12.2020 •

Mathematics

Last thing I'm gonna need help with

Last thing I'm gonna need help with

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Ответ:

bloodry

20.02.2021

$27\pi$

Step-by-step explanation:

The given curve is F(x,y,z)=xyi+5zk+7yk .

Let us parameterize C to obtain a vector equation for C.

We have x^2+y^2=81 , x^2+y^2=9^2 .

This is the equation of a circle( a cylinder in space) with radius, r=9 units.

The parametric equation of a circle is given by:

$x=r\cos t$ and $y=r\sin t$

This implies that:

$x=9\cos t$ and $y=9\sin t$

From, x+z=1 , we obtain, z=1-x , $\implies z=1-9\cos t$

A vector equation for C is;

$r(t)=9\cos ti+9\sin tj+(1-9\cos t)k$ for $0\le t\le 2\pi$

$\implies r'(t)=-9\sin ti+9\cos tj+9\sin tk$

Also, we have;

$F(r(t))=(81\cos t \sin t)i +5(1-9\cos t)j+(63\sin t)k$

Since C is oriented counterclockwise as viewed from above, we can apply the Stokes' Theorem.

Thus, by Stokes' Theorem;

$\int\limits \int\limits_Scurl F\cdot dS=\int\limits_CF\cdot dr$

$\int\limits \int\limits_Scurl F\cdot dS=\int\limits_0^{2\pi}F(r(t))\cdot r'(t)dt$

$=\int\limits_0^{2\pi}(-729 \cos t \sin^2t+45\cos t(1-9\cos t )+567 \sin^2t)dt=27\pi$

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