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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 ?
b) Solve for w x and draw its graph from x ± 0 to x ± 1. c) Solve for u x and draw its graph from x ± 0 to 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 ´ 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 ± ² 0. ∂ x 2 ∂ y 2 Show that u ² ∂ ∂ F y and s ² ∂ ∂ 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 Laplace’s equation div ± grad u ² 0? (i) v ± x , y ² ± x 2 , y 2 (ii) v ± x , y ² ± y 2 , x 2 (iii) v ± x , y ² ± x ± y , x − y c) (i) Find the solution to Laplace’s equation inside the unit circle r 2 ² x 2 ± y 2 ² 1 if the boundary condition on the circle is u ² u ± 2 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 2 1 , y ² 0....
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quiz2_18085f02 - c What can you say about u ± x y(what...

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