Engineering Mathematics II (010.141)

Final Examination

12 December, 2009

1. Answer the following questions.

(1) (10 points) Derive the continuous equation of a compressible fluid.

(2) (5 points) What is the physical meaning of the "compressible fluid" and the continuous equation?

2. (20 points) If the surface of the ball [eqn: r ^{2} =x ^{2} +y ^{2} +z ^{2} LEQ R ^{2}] is kept at temperature zero and the initial temperature in the ball is f(r). Find the temperature u(r,t) in the ball.

3. (15 points) Derive the Cauchy-Riemann equations and explain its relation with Laplace's equation briefly.

4. (15 points) Integrate [eqn: {z ^{2} +1} over {z ^{2} -1}] clockwise around the circle C of radius 1 with center at (1) z =1 (2) z = -1 (3) z = i

5. (10 points) Find Laurent series of [eqn: f(z)= {1} over {z ^{2} +4z+3}]

6. (20 points) Evaluate (1) [eqn: I= int _{0} ^{2 pi } {{d theta } over {1-2pcos theta +p ^{2}}} ```(0<p<1)]

(2) [eqn: I= int _{- INF } ^{INF } {{e ^{px}} over {1+e ^{x}} dx} ```(0<p<1)]

7. (5 points) Find the value of the Riemann zeta function [eqn: zeta (2)= sum _{n=1} ^{INF } {1} over {n ^{2}}] by expanding [eqn: f(x)=x ^{2} ,```- pi <x< pi] using

“The LORD is my shepherd, I shall not be in want.

He makes me lie down in green pastures, he leads me beside quiet waters,

he restores my soul. He guides me in paths of righteousness for his name's sake.

Even though I walk through the valley of the shadow of death, I will fear no evil, for you are with me; your rod and your staff, they comfort me.“ (Psalms 23:1-4)

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