Answered

Use Stokes' theorem to evaluate ∬S(∇×F)⋅dS where F(x,y,z)=−16yzi+16xzj+13(x2+y2)zk and S is the part of the paraboloid z=x2+y2 that lies inside the cylinder x2+y2=1, oriented upward.



Answer :

32π is the value of F by Stokes' theorem.

What exactly does Stokes Theorem mean?

In accordance with the Stoke's theorem, "the surface integral of the curl of a function over a surface bounded by a closed surface is equal to the line integral of the specific vector function around that surface."

Since F = −16yzi+16xzj+13(x2+y2)zk

     Here  x² + y²  = 1

put   x = 1 cos t     y = sin t     z = 1

F = - 16 Sin t i + 16cos t j + 13 . 1 k

       = -16 Sin t i + 16cos t j + 13k

C : Y(t) =  < x,y,z >

            = < cost , sint , 1 >

    dr = < -sint , cost , 0 >

 we know that stokes' theorem

         ∫∫s = Curl F. ds = ∫c F . dr

   ∫∫s curl F . ds = ∫₀²π < -16 sint , 16cost , 13> < - sint , cost , 0 >dt

                          = ∫₀²π ( 16 sin² t + 16 cos² t + 0 ) dt

                       = ∫₀²π 16( sin² t + cos² t ) dt

                        =  ∫₀²π 16 . 1  dt

                         = 16 [ t ]₀²π  = 16 * 2π  ⇒32π

Learn more about Stokes' theorem

brainly.com/question/29751072

#SPJ4

Other Questions