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18.085 Computational Science and Engineering I
Fall 2008
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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 cŸx du x− 3
4
dx
dx
uŸ0 0
wŸ1 0
Suppose that
cŸx 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 wŸx and draw its graph from x 0 to x 1.
c) Solve for uŸx and draw its graph from x 0 to x 1.
Problem 2 (34 points)
a)
(i) Find the real part uŸx, y and the imaginary part sŸx, y of
1
fŸz 1z x iy
(ii) Also find uŸr, 2 and sŸr, 2 for the same function expressed in polar coordinates:
fŸz 1z 1i2
re
b) Draw the equipotential curve uŸx, y 12 and the streamline sŸx, 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 uŸx, y (what condition does it satisfy) along the line s 12 ?
Problem 3 (33 points)
a). Suppose that the Laplacian of FŸx, 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 uŸx, y and what is that function?
Does uŸx, y solve Laplace’s equation divŸgrad u 0?
(i) vŸx, y Ÿx 2 , y 2
(ii) vŸx, y Ÿy 2 , x 2
(iii) vŸx, y Ÿx 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 Ÿ2 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.