MATH 2700: CALCULUS 3 SPRING 2020 P2 ASSIGNMENT # 06 (40 points.) NAME: DUE: Monday May 4th, 2020 DIRECTIONS: Your solutions must be neat and complete to receive full credit. ~ (x, y , z) = y , x, z 2 . Let Q be the solid bounded by z = 4 and z = x 2 + y 2 with boundary S. 1. Let F (a) Sketch Q and the projection of Q in the xy -plane. ~ across S by computing: (b) Use the Divergence Theorem to find the outward flux of F ZZZ ~ dV F Q HINT: This is best evaluated using cylindrical coordinates. 15 ~ (x, y , z) = h3z, 2y , xi. Let S be the portion of x + 3y + 2z = 12 which lies in the first octant. 2. Let F Let C be the boundary of S oriented by the upward pointing normal. (a) Sketch S and its projection into the xy -plane. (b) Use Stokes's Theorem to find the work done moving a particle along C by computing: ZZ ~)N up dS ( F S HINT: Feel free to use properties of double integrals to simplify your computation! 15 3. (a) Using the \u000f definitions of lim ~r (t) = ~L and ta lim f (x, y ) = L (x,y )(a,b) as guides, formulate the \u000f definition of lim ~ (x, y ) = ~L F (x,y )(a,b) (b) Using limit definitions of: '~r is continuous at a' and 'f (x, y ) is continuous at (a, b)' as guides, formulate ~ (x, y ) is continuous at (a, b).' the limit definition of: 'F (c) Using the limit definitions of ~r 0 (t) and fx (x, y ) as guides, formulate the limit definition of ~ru (u, v ) for a vector valued function ~r (u, v ). 10