4. Let C be a path in R from the point (1, 0, 2) to the point (-2, 1, 0), and let f(x, y, z) = xz + 3y. Use the fundamental theorem of conservative vector fields to evaluate Vf(x, y, z) . di. 5. Let C be the triangle in R2 with vertices (0, 0), (0, 2) and (2, 0), oriented counterclockwise. Let F(x, y) = (cos(y), e"). (a) Use Green's Theorem to write a double integral that is equivalent to the line integral O F. r(1). C (b) Compute the partial derivatives needed for the surface integral. (c) Evaluate the surface integral.7". Let 5' he the portion of the surface a = x2 + y: lying above the rectangle where '05): 5 lano'lillii}1 51.Letf{x,y)=ltixy. (a) Write a parametrization (II-{L y) for S. (h) Compute the length of a normal vector to 5' at an arbitrary point (x. y). (c) Compute the scalar surface integral JJ x, y) d3. 3 8. Let |F[x, _v, 2:) = {29, 2, x}. and let S be the upper hemisphere of the unit sphere centered at the origin, with outwardpointing normal vectors. (a) Write a parametrization (MS. rl} for S in spherical coordinates. including the intervals for H and if). (h) Compute the normal vector to S at an arbitraryr point {1, y]. (c) Compute the vector surface integral J[ IFLL _v, z] - d3 5 9. Let F(x, y, z) = (2xy, x, y + z) and let C be the square with vertices (0, 0, 0), (1, 0, 0), (1, 1, 0), and (0, 1, 0), oriented counterclockwise. (a) Write a parametrization O(x, y) for the planar region enclosed by C. (b) Compute the curl V XF. (c) Use Stokes' Theorem to write a vector surface integral equivalent to the vector line integral o F. dr, where r(t) is some parametrization of C. C111}. Let x, _v, z) = (xv, xz. gig) and let S be the surface of the region enclosed by 12+}12 = 1.5; =U,and:; = 2. (a) Compute the divergence V . IF. (b) U se the Divergence Theorem to write a triple integral equivalent to the vector surface integral JI [F(x,y,z] - d8. 5' (c) Evaluate your integral front part (b). 1. Determine whether or not each vector field is conservative. (a) F(x, y) = (xy, 4xy) (b) F(x, y) = (y', 3xy?) 2. Let C be the upper half of the circle x2 + y = 9 and let f(x, y) = 3 + xy. (a) Parametrizationunction r(t). (b) Compute ||7'(1)II. (c) Compute the scalar line integral f(x, y) ds. 3. Define the vector field F(x, y) = (x + y, y), and let C be the path in the plane parametrization) with 0