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MIT OpenCourseWare http://ocw.mit.edu 18.085 Computational Science and Engineering I Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. Quiz 2 18.085 Name_____________ Professor Strang November 4, 2002 Grading 1 2 3 −−−−−− Problem 1 (33 points) This question is about a fixed-free hanging bar (made of 2 materials) with a point load at x − d cx du x− 3 4 dx dx u0 0 w1 0 Suppose that cx a) Which of u, du dx 1, x 4, x , and w c du have jumps at (i) x dx 1 2 1 2 1 2 and (ii) x 3 4 ? 3 4 : b) Solve for wx and draw its graph from x 0 to x 1. c) Solve for ux and draw its graph from x 0 to x 1. Problem 2 (34 points) a) (i) Find the real part ux, y and the imaginary part sx, y of 1 fz 1z x iy (ii) Also find ur, 2 and sr, 2 for the same function expressed in polar coordinates: fz 1z 1i2 re b) Draw the equipotential curve ux, y 12 and the streamline sx, y 12 . (I suggest to use x-y coordinates and "clear out" denominators.) What shapes are these two curves? c) What can you say about ux, y (what condition does it satisfy) along the line s 12 ? Problem 3 (33 points) a). Suppose that the Laplacian of Fx, y is zero: ∂ 2 F ∂ 2 F 0. ∂x 2 ∂y 2 Show that u ∂F ∂y and s ∂F ∂x satisfy the Cauchy-Riemann equations. b). Which of these vector fields are gradients of some function ux, y and what is that function? Does ux, y solve Laplace’s equation divgrad u 0? (i) vx, y x 2 , y 2 (ii) vx, y y 2 , x 2 (iii) vx, y x y, x − y c) (i) Find the solution to Laplace’s equation inside the unit circle r 2 x 2 y 2 1 if the boundary condition on the circle is u u 0 2 12 cos 2 cos 22. (OK to use polar coordinates.) (ii) Find the numerical value of the solution u at at the center and at the point x 12 , y 0.