TEXAS A&M UNIVERSITY DEPARTMENT OF MATHEMATICS MATH 251-511 Final exam version A, 12 Dec 2007 Name: Points: /90 In all questions, no analytical work — no points. 1. Given two planes with equations x + 2y + z = 1 and x − 2y + 3z = 3, (a) Show that the planes are orthogonal. (b) Find the plane that is orthogonal to the given ones and passes through the point (1, −1, 2). (15 points) 2. Find all critical points of the function f (x, y) = 3 + x2 + y 2 + xy 2 . Determine the type of each point (minimum / maximum / saddle). (15 points) 3. A solid E is bounded by the cylinder x2 + y 2 = 1 (from the sides), the plane z = 4 (from the top) and the paraboloid z = 1 − x2 − y 2 (from the bottom). Sketch the solid. If the density at any point is proportional to its distance from the axis of the cylinder, find the mass of E. (15 points) 4. Evaluate Z (y 2 + 1) dx + 2xy dy, C if (a) C is the straight line from (−1, 0) to (1, 0). (b) C is the arc of the circle x2 + y 2 = 1 going counter-clockwise from (−1, 0) to (1, 0). (15 points) 5. Calculate the mass of the (surface of the) hemisphere x2 + y 2 + z 2 = 1, z ≥ 0, if the density function is ρ(x, y, z) = 1 − z 2 . (15 points) 6. Calculate the flux of F(x, y, z) = hx3 + y cos y, y 3 + x sin z, 3i across the surface of the solid p p bounded by the hemispheres z = 4 − x2 − y 2 , z = 1 − x2 − y 2 and the plane z = 0. The surface is oriented outward. (15 points) 7. Bonus question +10% (no partial credit): Cheburashka and Krokodil Gena are showing off their knowledge of vector calculus. Cheburashka is claiming that curl div F = 0, (1) while Gena argues that, on the contrary, div curl F = 0. (2) Who is correct? Prove the correct formula and explain why the other is wrong. Here F is a smooth 3-dimensional field.