ENGR 251 Final: ENGR 251 Final Exam 2006 Winter
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Instructors: dafni, dryanov, enolskii, keviczky, kisilevsky, korotkin, shnirelman. Compute the curvature (t) of the curve c de ned by. Find points on the surface x2 + 3y2 + 4z 2 2xy = 16 at which the tangent plane is parallel to the yz plane. Find the direction in which the function below increases most rapidly at the indicated point. Find also the maximum rate of increase. f (x, y) = e2x sin(2y) , Y2 u (t, x, y) = 0. describes the propagation of small waves on an elastic membrane. C (x + 2y)dx + (2x y)dy where c is the contour indicated in gure starting at ( 4, 1) and ending at ( 3, 1). x. Using the appropriate theorem (which you must state), compute the ux of the curl for the vector eld across the upper hemisphere. ~f = yi xj + z cos(z 3 + ln(1 + x2))k.
