1. Let S be the triangular surface with corners at (1, 0, 0), (0, 1, 0) and (0, 0, 1). Let F be given by F = zxi + xyj + yzk Compute V X F . nds; you can define which of the two possible normal vectors you want to use for yourself. Also compute the corresponding line integral that appears in Stokes' theorem directly, and show this to confirm that Stokes' theorem does hold in this case. 2. Let C be the curve parameterized by x = cos(t), y = sin(t), z = at. Compute the length of the curve from t = 0 to t = 27. Repeat for the curve x = cos(t), y = sin(t), 2 = 2+3/2 3. Let B(x, y, z) be an arbitrary magnetic field. For a charged particle with charge q travelling at velocity v, the force on the particle is F =qBX v. Note that this is different from gravitational and electrostatic fields, for which the force at a point is simply proportional to the field itself: F here is not simply a standard vector field, since its magnitude and direction don't depend just on the position (that would be true for B but not F). Consequently, we cannot talk Instead, F depends on the velocity v at which the particle travels, and that is not a vector field but dependent on the motion of the particle v, which is not defined for an arbitrary position but instead is only defined along the trajectory of the particle.about a curl of F, or about F being a conservative vector field (it isn't a vector field! - different particles at the same point will be subject to different forces if they travel at different velocities). Nonetheless, we can show that work done by a magnetic field is always zero. Let C be an arbitrary path followed by the particle. C does not need to be a closed loop. Show that if F is given in terms of B and velocity v as above, then the amount of work ( F . dr done in moving the particle along that path is always zero. Hint: parameterize the curve C' in terms of time elapsed t when computing F . dr