Given that curl f = 2yi – 2zj + 3k, find the surface integral of the normal component of curl f (not f) over (a) the open hemispherical surface x2 + y2 + z2 = 9, z > 0. (b) the sphere x2 + y2 + z2 = 9. (in both parts, you should be able to write the answer down by inspection.)

Use Stokes' theorem for both parts, which equates the surface integral of the curl to the line integral along the surface's boundary.

a. The boundary of the hemisphere is the circle in the plane , where the curl is . Green's theorem applies here, so that

which means the value of the line integral is 3 times the area of the circle, or .

b. The closed sphere has no boundary, so by Stokes' theorem the integral is 0.

it would open into a square depending on the type of envelope i’m not sure about the second part

i only know the sequence of arithmetic it adds the first is 5+4 then it goes up as it adds to then it is 9+5 then 14+6 then to 20+7 and so on. 5,9,14,20,27,35.

step-by-step explanation: