Evaluate the surface integral s f · ds for the given vector field f and the oriented surface s. in other words, find the flux of f across s. for closed surfaces, use the positive (outward) orientation. f(x, y, z) = y i − x j + z2 k s is the helicoid (with upward orientation) with vector equation r(u, v) = u cos v i + u sin v j + v k, 0 ≤ u ≤ 2, 0 ≤ v ≤ 5π

The magnetic field inside a superconducting solenoid is 4.50 t. the solenoid has an inner diameter of 6.20 cm and a length of 26.0 cm. determine (a) the magnetic energy density in the field and (b) the energy stored in the magnetic field within the solenoid.

Atank has a shape of a cone with a radius at the top of 2 m and a height of 5 m. the tank also has a 1 m spout at the top of the tank. the tank is filled with water up to a height of 2 m. find the work needed to pump all the water out the top of the spout. (use 9.8 m/s2 for g and the fact that the density of water is 1000 kg/m3.)

Assume that two of the electrons at the negative terminal have attached themselves to a nearby neutral atom. there is now a negative ion with a charge -2e at this terminal. what are the electric potential and electric potential energy of the negative ion relative to the electron? the electric potential and the electric potential energy are both twice as much. the electric potential is twice as much and the electric potential energy is the same. the electric potential is the same and the electric potential energy is twice as much. the electric potential and the electric potential energy are both the same. the electric potential is the same and the electric potential energy is increased by the mass ratio of the oxygen ion to the electron. the electric potential is twice as much and the electric potential energy is increased by the mass ratio of the oxygen ion to the electron.