# SM311o - Second Exam

1. Let v = áx2 -y2, xyz, [Ö(x2 + y2)]ñ.
1. Determine the divergence of v. Find a point P = (x, y, z) at which the flow is incompressible.
2. Determine the vorticity of v. Is this vector field irrotational at P = (1, 0, 1)?
3. Write a Mathematica program that computes the divergence of the above vector.
2. Let v = áf(x, y, z), g(x, y, z), h(x, y, z)ñ be a velocity field. Let w be the vorticity of this velocity field. Prove that div w = 0.
3. Let v = áx-y, y-z, z-xñ. Let C be the ellipse given by x2 + 3y2 = 4. Determine the circulation of v around C.
4. Let v be the velocity field of a two dimensional flow. Suppose that v is incompressible and irrotational. Show that the stream function of this flow must satisfy
 ¶2 y ¶ x2 + ¶2 y ¶ y2 = 0.
5. Find all eigenvalue-eigenfunction pairs of the Laplace operator -D in the square D = [0,a]×[0,b] subject to the following boundary conditions
 u(x,0) = u(0,y) = 0, ¶ u ¶ x (a,y) = ¶ u ¶ y (x,b) = 0.
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