Question

Let M be the capped cylindrical surface which is the union of two surfaces, a cylinder given by x^2 + y^2 = 16, 0 ? z ? 1, and a hemispherical cap defined by x^2 + y^2 + (z - 1)^2 = 16, z ? 1. For the vector field F = (zx + z^2y + 6y, z^3yx + 6x, z^4x^2), compute ?_M(? × F) ? dS in any way you like. ?_M(? × F) ? dS = 3pi

          Let M be the capped cylindrical surface which is the union of two surfaces, a cylinder given by x^2 + y^2 = 16, 0 ? z ? 1, and a hemispherical cap defined by x^2 + y^2 + (z - 1)^2 = 16, z ? 1. For the vector field F = (zx + z^2y + 6y, z^3yx + 6x, z^4x^2), compute ?_M(? × F) ? dS in any way you like.
?_M(? × F) ? dS = 3pi

Let M be the capped cylindrical surface which is the union of two surfaces, a cylinder given by x^2 + y^2 = 16, 0 ? z ? 1, and a hemispherical cap defined by x^2 + y^2 + (z - 1)^2 = 16, z ? 1. For the vector field F = (zx + z^2y + 6y, z^3yx + 6x, z^4x^2), compute ?M(? × F) ? dS in any way you like.
?M(? × F) ? dS = 3pi

Added by Gabriel C.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Namya K

03/04/2022

Step 1

For the cylinder, we can use the parametrization: $$ \vec{r}_1(u,v) = (4\cos{u}, 4\sin{u}, v), \quad 0 \le u \le 2\pi, \quad 0 \le v \le 1 $$ For the hemispherical cap, we can use the parametrization: $$ \vec{r}_2(u,v) = (4\cos{u}\sin{v}, 4\sin{u}\sin{v}, 1 + Show more…

Show all steps

Thanks for your feedback!

Let M be the capped cylindrical surface which is the union of two surfaces, a cylinder given by x^2 + y^2 = 16, 0 <= z <= 1, and a hemispherical cap defined by x^2 + y^2 + (z - 1)^2 = 16, z >= 1. For the vector field F = (zx + z^2y + 6y, z^3yx + 6x, z^4x^2), compute ∬_M(∇ × F) ⋅ dS in any way you like. ∬_M(∇ × F) ⋅ dS = 3pi