Let Find the outward flux of the vector field F across the surface of the region bounded above by the sphere by x² + y² + z² = 4 and below by the plane z = 0.

Question

Quizplus · Accepted Answer

We can use the Divergence Theorem to calculate the outward flux of the vector field F across the given surface. The divergence of the vector field is:

div(F) = 2x + 2y + 2z

The triple integral of the divergence over the region R bounded above by the sphere and below by the plane is:

int_int_int_R div(F) dV = int_0^(2*pi) int_0^pi int_0^2 (2r^2 sin(theta)) dr dtheta dphi

Using spherical coordinates, where x = r sin(theta) cos(phi), y = r sin(theta) sin(phi), and z = r cos(theta), we can simplify this integral to:

int_0^(2*pi) int_0^pi int_0^2 (2r^2 sin(theta)) dr dtheta dphi = 2 int_0^(2*pi) int_0^pi int_0^2 (r^2 sin(theta)) dr dtheta dphi

= 2 (int_0^(2*pi) dphi) (int_0^pi sin(theta) dtheta) (int_0^2 r^2 dr)

= 2 (2*pi) (2) (8/3)

= 85.33

Therefore, the outward flux of F across the surface is:

int_int_S F dot n dS = int_int_R div(F) dV = 85.33

The closest choice is B) 11ead0bc_8d84_efa3_99a0_6798e8040299_TB6988_11.

Let Find the Outward Flux of the Vector Field