ME760 Engineering Analysis I

Homework Set 5 due: Friday Nov. 9, 2015 1. Verify that the contour integral R C[2xy2 dx+2x2y dy+dz] is independent of the path. Evaluate this integral between the points (0, 0, 0) and (a, b, c). 2. Given the parametric form of a cone r(u, v) = [u cos v, u sin v, cu] (a) find an explicit represen- tation of the form z = f(x, y), (b) find and identify the parameter curves defined as u = const and v = const, and (c) find a normal vector N to the conical surface. 3. In class we discussed surface integrals without regard to orientation. By reparameterizing the surface integral could be written as I = ZZ S G(r)dS = ZZ R G(r(u, v))|N(u, v)|du dv (a) Consider the case with G = z and the surface S is the hemisphere x2 + y2 + z2 = 9 with z 0. Use polar coordinates and evaluate the right-hand side of the above result. (b) The surface S is also given explicitly by z = f(x, y) = p 9 - x2 - y2. For such cases the surface integral can be rewritten as ZZ S G(r)dA = ZZ R G(x, y, f(x, y)) s 1 + @f @x 2 + @f @y 2 dx dy. Evaluate the right-hand side of this result. 4. Evaluate RR S F•ˆndA using the divergence (Gauss’) theorem when (a) F = [x3, y3, z3] and the surface S is the sphere x2 + y2 + z2 = 9, and (b) when F = [9x, y cosh2 x,-z sinh2 x] and S is the ellipsoid 4x2 + y2 + 9z2 = 36. 5. Consider the vector function F = [ez, ez sin y, ez cos y] and the surface S : z = y2, 0 x 4, 0 y 2. Stoke’s theorem states that ZZ S (r × F)•ˆndS = I C F•dr. (a) Evaluate the left-hand side of this result, and (b) evaluate the right-hand side.