Question

Let M be the capped cylindrical surface which is the union of two surfaces, a cylinder given by ^x2+y²=49,0≤z≤1, and a hemispherical cap defined by x²+y²+(z−1)²=49,z≥1. For the vector field F=(zx+z²y+7y,z³yx+3x,z⁴x²), compute ∬M(∇×F)⋅dS in any way you like. ∬M(∇×F)⋅dS=

Answer 1

By applying Stokes theorem, the value of the integral is equal to -52π.

How to use the Stokes theorem?

First of all, we would use Stokes theorem to convert the integral as follows: int s (Delta*F) .mds= int c Fdr

where:

c represents the boundary circle on the xy-plane.

Note: The bounding curve (x ^ 2 + y ^ 2 = 49) is a circle located at the origin with a radius of 7.

Thus, the parametric equation for the circle is given by:

r(t)=7 cos t hat i + 7sin tj t is from 0 to 2Pi

Differentiating r(t) with respect to t, we have: r^ prime (t)=-7 sin t hat i + 7cos tj .

Next, we would compute F(r(t)) :

F(r(t))=8(7 sint) hat i + 5(7cos t) * j z = 0 delta*z = 0'

By taking the dot product, we have: F(r(t)) * r' * (t) = - 392sin^2 t + 245cos^2 t

Note: The double-angle formula for cos is cos(2t) = 2cos^2 t - 1

Now, we can integrate from 0 to 2π: integrate (- 392sin^2 t + 245cos^2 t) dt from 0 to 2pi

integrate (- 392/7 * (1 - cos 2t) + 245/7 * (1 + cos 2t)) dt from 0 to 2pi

-392 7 int 0 ^ 2 pi (1 - cos 2t) + 245/7 * integrate (1 + cos 2t) dt from 0 to 2pi

int s ( Delta* F). m * (ds = - 56[2pi + 0] + 35[2pi + 0])

int s ( Delta* F). m * (ds = - 112pi + 70pi)

int s ( Delta* F). m * ds = - 52pi iint m( nabla* f)* ds = - 52pi .