Math 241 Final Exam Part one 1. (a) State the divergence theorem. (b) Using any method you wish, evaluate !! S ! F! · dS where F! = !3x, xy, 2xz" and S is the surface of the solid cube [0, 1] × [0, 1] × [0, 1]. 2. Which of the following two vector fields is conservative? F!1 = !yex , ex + 1" F!2 = !y 2 , x" What is the potential function for the conservative vector field? 3. Below are picture four oriented curves, C1 , C2 , C3 , C4 in a vector field F! . For each of these curves, state whether ! F! · d!r C is positive, negative or zero. C2 C1 C3 C4 4. Draw three pictures of a two-dimensional vector field: one with positive divergence at (0, 0), one with negative divergence at (0, 0) and one with zero divergence at (0, 0). Explain why your pictures have the required property. " ! where F = !x, y, y" and S is the triangle with vertices (1, 0, 0), (0, 1, 0), (0, 0, 1). 5. Calculate S F! ·dS, 6. (a) State Green’s theorem. 1 (b) Using any method you wish, evaluate ! ey dx + 2xey dy C where C is the square with sides x = 0, x = 1 and y = 0, y = 1. Part Two 1. Let f be a function and let !a, !b, !c, d! be vector fields. State whether each expression is meaningful. If so, state whether it is a function or a vector field. ! (a) (!a × !b) · (!c × d) (b) div(!a · !b) (c) div(curl !a) (d) curl(curl(curl !a)) (e) (!a · !b) × !c (f) ∇f × curl !a (g) ∇(!a · !b) (h) ∇(curl !a) (i) ∇(∇f ) (j) div(∇f ) 2. Find the maximum and minimum values of f (x, y) = 4x + 6y subject to the constraint x2 + y 2 = 13. 3. Let f (x, y, z) = x2 − y 2 − z 2 . At (1, 1, 1), what is the direction in which f increases fastest? Which direction does f decrease fastest? 4. Let f (α, β) = αeα+β + sin β. Find fα , fβ , fαα , fββ , fαβ . # 5. What is the average value of f (x, y) = x2 + y 2 on the disk of radius a, x2 + y 2 ≤ a2 ? 6. What is the equation of the plane through the three points (1, 1, 1), (1, 1, 2), (0, 0, −1)? 2