where S is the helicoid:
r(u, v) = ucos(v)i + usin(v)j + vk,
with 0 &le u &le 4, 0 &le v &le 5&pi
Let S be the part of the plane
4x + 4y + z = 4 which lies in the first octant,
oriented upward. Find the flux of the vector field
F = 2i + 2j + 3k
across the surface S.
A fluid has density 3 kg/m^3 and flows in a velocity field
v = -yi + xj + 4zk
where x, y , and z are measured in
meters and the components of v in meters per second.
Find the rate of flow outward through the sphere
Let M be the closed surface that consists of the hemisphere
and its base
Let E be the electric field defined by
E = (2x, 2y, 2z). Find the electric flux across M.
Write the integral over the hemisphere using spherical coordinates,
and use the outward pointing normal.