Engineering Mathematics II (010.141)

Midterm Examination

23 October, 2009

1. Answer the following questions.

(1) (5 points) What is the gradient? How is it related to directional derivatives?

(2) (5 points) What is the definition of the divergence? Its physical meaning?

Its relation to the Laplacian?

(3) (5 points) What is the definition of the curl? Its physical meaning?

(4) (5 points) What is scalar triple products? Its geometrical meaning?

(5) (5 points) What is the meaning of the following equations?

[eqn: NABLA TIMES E=- {Partial B} over {Partial t}# NABLA TIMES B= mu _{0} J# NABLA BULLET E= rho / epsilon _{0}# NABLA BULLET B=0]

2. Answer the following questions.

(1) (5 points) What is the meaning of "compressible fluid"?

(2) (10 points) Derive the continuous equation of a compressible fluid.

[eqn: {Partial rho } over {Partial t} + NABLA BULLET ( GAMMA )=0]

ρ: density (mass per unit volume)

Γ = ρv: flux

(3) (5 points) What is the physical meaning of the continuous equation?

3. Prove the following formulae.

(1) (5 points) b×(c×d) = (b⦁d)c - (b⦁c)d

(2) (5 points) Lagrange's identity: (a×b)⦁(c×d) = (a⦁c)(b⦁d) - (a⦁d)(b⦁c)

4. (1) (5 points) What is the Fourier integral?

(2) (5 points) Derive the complex form of the Fourier integral and derive the general form of the Fourier transform from it.

(3) (5 points) Find the Fourier transform of [eqn: xe ^{-x ^{2}}], using Ƒ{[eqn: e ^{-ax ^{2}}]} = [eqn: {1} over {sqrt {2a}} e ^{-w ^{2} /4a}] (a>0)

5. (10 points) Evaluate [eqn: oint _{C} ^{} {F BULLET r prime ds} ,``F=(x ^{2} +y ^{2} ) ^{-1} [-y,`x],``C:x ^{2} +y ^{2} =1,`z=0], oriented clockwise. Why can Stokes's theorem not be applied? What result would it give?

7. Find the following.

(1) (10 points) Find representation and surface area of a torus.

A torus surface S is obtained by rotating a circle C about a straight line L in space so that C does not intersect or touch L but its plane always passes through L. If L is the z-axis and C has radius b and its center has distance a (>b) from L.

(2) (10 points) Evaluate [eqn: int _{0} ^{INF } {{coswx} over {k ^{2} +w ^{2}}} dw+ int _{0} ^{INF } {{wsinwx} over {k ^{2} +w ^{2}}} dw]