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Math 251 Week in Review Sections 14.5-14.6 1. Find a potential function or show the vector field is not conservative. a) F ( x , y , z ) x i e sin z j e cos z k y y b) F ( x , y , z ) xz i xy j yz k c) F ( x , y , z ) ( 2 xy z y ) i (3 x y z 2 yz 3 xy ) j ( x y y ) k 2. F xy 3 2 i yz 2 3 2 2 2 2 3 2 j zx k Does there exist a vector field G for which 2 F curl (G ) ? Hint: What is div( F )? 3. a) Show that any vector field of the form f ( x ) i g ( y ) j h ( z ) k is irrotational. b) Show any vector field of the form f ( y , z ) i g ( x , z ) j h ( x , y ) k is incompressible. 4. Parameterize the.upper half of the ellipsoid 4 x 9 y z 36 : 2 2 2 a) using x and y as parameters. b) using a polar form on 2x and 3y. c) using a spherical form with φ and θ as parameters. 5. Find the surface area of the paraboloid z x y 2 6. Find the surface area of the cylinder x z 2 2 2 0 z 9. a 2 that lies inside y 2 z 2 a 2 . 7. Find an equation of the tangent plane, at the point (1,0,1), to the surface parameterized as x (u , v ) u 2 y (u , v ) u v 2 z v2 .