c Dr Oksana Shatalov, Spring 2013 1 Spring 2013 Math 251 Week in Review 10 courtesy: Oksana Shatalov (covering Sections 14.5-14.6 ) 14.5: Curl and Divergence Key Points i ∂ • curlF = ∇ × F = ∂x P j ∂ ∂y Q k ∂ ∂z R • If F is conservative, then curlF = 0. • divF = ∇ · F • div curl F = 0 1. Given the vector field F = zi + 2yzj + (x + y 2 )k. (a) Find the divergence of the field. (b) Find the curl of the field. (c) Is the given field conservative? If it is, find a potential function. c Dr Oksana Shatalov, Spring 2013 (d) Compute R C 2 z dx + 2yz dy + (x + y 2 ) dz where C is the positively oriented curve y 2 + z 2 = 4, x = 2013 R (e) Compute C z dx + 2yz dy + (x + y 2 ) dz where C consists of the three line segments: from (0, 0, 0) to (2013, 0, 0), from (2013, 0, 0) to (2, 3, 1), and from (2, 3, 1) to (1, 1, 1). 2. Is there a vector field such that curlF = −2xi + 3yzj − xz 2 k? 14.6: Parametric surfaces and their areas Key Points • Parametric representation of surfaces: x = x(u, v), y = y(u, v), z = z(u, v), (u, v) ∈ D. • The graph of a vector-function of two variables, r(u, v) = x(u, v)i + y(u, v)j + z(u, v)k, surface. (u, v) ∈ D, is a • A normal vector to the tangent plane to a parametric surface at point (x0 , y0 , z0 ) = (x(u0 , v0 ), y(u0 , v0 ), z(u0 , v0 )) is N(u0 , v0 ) = ru (u0 , v0 ) × rv (u0 , v0 ). RR RR • Surface area: A(S) = S dS = D |ru × rv |dA. 3. Identify the surface x = √ u cos v, y = 5u, z = √ u sin v. c Dr Oksana Shatalov, Spring 2013 4. Identify the surface which is the graph of the vector-function r(u, v) = hu + v, u − v, ui. 5. Find a parametric representation of the following surfaces: (a) x + 2y + 3z = 0; (b) the portion of the plane x + 2y + 3z = 0 in the first octant; (c) the portion of the plane x + 2y + 3z = 0 inside the cylinder x2 + y 2 = 9; (d) z + zx2 − y = 0; (e) y = x2 ; 3 c Dr Oksana Shatalov, Spring 2013 4 (f) the portion of the cylinder x2 + z 2 = 25 that extends between the planes y = −1 and y = 15 6. Find an equation of the plane tangent to the surface x = u, y = 2v, z = u2 + v 2 at the point (1, 4, 5). 7. Given a sphere of radius 2 centered at the origin, find an equation for the plane tangent to √ it at the point (1, 1, 2) considering the sphere as p (a) the graph of g(x, y) = 4 − x2 − y 2 (Note that the graph of g(x, y) is the upper halfsphere ); c Dr Oksana Shatalov, Spring 2013 5 (b) a level surface of f (x, y, z) = x2 + y 2 + z 2 ; (c) a surface parametrized by the spherical coordinates. 8. Find the area of the part of the cylinder x2 + z 2 = 1 which lies between the planes y = 0 and x + y + z = 4. c Dr Oksana Shatalov, Spring 2013 6 9. Find the area of the portion of the cone x2 = y 2 + z 2 between the planes x = 0 and x = 2.