Suppose F 2.xz + 3y²) ay + (4yz²) az. (a) Calculate fLL F dS, where S is the shaded surface in Figure 1. (b) Calculate SF· dl, where C is the A → B → C → D → A closed path in Figure 1. (c) Based on your results for parts (a) and (b), what named theorem do you think is being satisfied here, if any? az C С (0,1,1) D (0,0,0) A ay В ax Figure 1: Figure for Problem 1.

Let F = (3z + 3x²) ¿+ (6y + 4z + 4 sin(y²)) 3+ (3x + 4y + 6e²²) k (a) Find curl F. curl F - (b) What does your answer to part (a) tell you about SF. dr where C' is the circle (x − 15)² + (y – 10)² = 1 in the xy-plane, oriented clockwise? ScF. dr = 0 (c) If C is any closed curve, what can you say about ScF. dr? SoF. dr = 0 (d) Now let C be the half circle - (x − 15)² + (y − 10)² = 1 in the xy-plane with y > 10, traversed from (16, 10) to (14, 10). Find SF. dr by using your result from (c) and considering C plus the line segment connecting the endpoints of C. -0 Lc F. dr =

5. Find the equation of the plane tangent to the surface x² + y ² at (3,1, 10). Z=

33. Figure 14.13 shows the saddle-shaped surface z = S(x, y). (a) What is the sign of f,(0, 5)? (b) What is the sign of f,(0,5)? (0, 5, 3) y Figure 14.13