# SM311o - Second Exam

- Let
**v**= áx^{2}-y^{2}, xyz, [Ö(x^{2}+ y^{2})]ñ.- Determine the divergence of
**v**. Find a point P = (x, y, z) at which the flow is incompressible. - Determine the vorticity of
**v**. Is this vector field irrotational at P = (1, 0, 1)? - Write a
*Mathematica*program that computes the divergence of the above vector.

- Determine the divergence of
- 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. - Let
**v**= áx-y, y-z, z-xñ. Let C be the ellipse given by x^{2}+ 3y^{2}= 4. Determine the circulation of**v**around C. - 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

¶ x^{2}+ ¶ ^{2}y

¶ y^{2}= 0. - 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. (1)

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