1. Prove the given properties for the vector fields F and G, scalar c, and scalar function f. Assume that the required partial derivatives are continuous. (a) V . (cF) = cV . F. . ( D+ 1 ) . A ( q ) (c) VX (cF) = c(V x F). ( d ) V x ( F + G) = V x F +V xG. ( e) V . ( f F ) = fV . F + VS . F. (f ) Vx ( Vf ) = 0. (g ) V . ( V x F ) = 0. 2. Let R = (x, y, z) and r = |R||. Find the work done in moving an object from a distance a to a distance b in the force field F = R/73. 3. Use Green's Theorem to evaluate d xy?da + (x y + 2x) dy where C is counter-clockwise around the JC square with vertices (ta, ta) with a > 0. 4. Find the area of the cap cut from the sphere x2 + y2 + 22 = 2 by the cone z = va2 + y2. 5. (a) Use Stokes' theorem to calculate the circulation of the vector field F around the curve C. That is, find fo F . dr when F = (y? + 22) i+ (x2 + 2?) j+ (x2 + y?) k and C is the boundary of the triangle cut from the plane a + y + 2 = 1 by the first octant. The curve C is oriented counterclockwise when viewed from above. (b) Use Stokes' Theorem to calculate V x FdS when F = (3y, 5 - 2x, 22 -2). S is parameterized by r(r, 0) = (r cos0, rsin 0,5 -r) with 0