quiz2_18085f02

quiz2_18085f02 - c) What can you say about u x , y (what...

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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 .
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18.085 Quiz 2 Professor Strang Name_____________ November 4, 2002 Grading 1 2 3 −−−−−− 3 : 4 Problem 1 (33 points) This question is about a fixed-free hanging bar (made of 2 materials) with a point load at x ± d c ± x du x 3 ± - 4 dx dx u ± 0 ± 0 w ± 1 ± 0 Suppose that 1, x ² 1 2 c ± x ± 4, x ³ 1 2 du du 1 3 a) Which of u , dx ,and w ± c dx have jumps at (i) x ± 2 and (ii) x ± 4 ?
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b) Solve for w ± x and draw its graph from x ± 0to x ± 1. c) Solve for u ± x and draw its graph from x ± 0to x ± 1.
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Problem 2 (34 points) a) (i) Find the real part u ± x , y and the imaginary part s ± x , y of f ± z ± 1 z ± x ² 1 iy (ii) Also find u ± r , 2 and s ± r , 2 for the same function expressed in polar coordinates: f ± z ± 1 z ± 1 i 2 re b) Draw the equipotential curve u ± x , y ± 1 2 and the streamline s ± x , y ± 1 2 . (I suggest to use x - y coordinates and "clear out" denominators.) What shapes are these two curves?
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Unformatted text preview: c) What can you say about u x , y (what condition does it satisfy) along the line s 2 1 ? Problem 3 (33 points) a). Suppose that the Laplacian of F x , 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 u x , y and what is that function? Does u x , y solve Laplaces equation div grad u 0? (i) v x , y x 2 , y 2 (ii) v x , y y 2 , x 2 (iii) v x , y x y , x y c) (i) Find the solution to Laplaces equation inside the unit circle r 2 x 2 y 2 1 if the boundary condition on the circle is u u 2 2 1 cos 2 cos2 2 . (OK to use polar coordinates.) (ii) Find the numerical value of the solution u at at the center and at the point x 2 1 , y 0....
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This note was uploaded on 04/12/2011 for the course MATH 18.085 taught by Professor Staff during the Fall '10 term at MIT.

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quiz2_18085f02 - c) What can you say about u x , y (what...

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