AMATH 231 ASSIGNMENT # 9 Stokes Theorem Fall 2014

Due Monday, November 24, 2014 at 2pm in box 7, slot 11 (A-M) and 12 (N-Z), locatedacross from MC4066. Late assignments or assignments submitted to the incorrect dropboxwill receive a grade of zero. Write your solutions clearly and concisely. Marks will bededucted for poor presentation and incorrect notation.

1. Compute the circulationC~F d~x directly and then use Stokes theorem to verify your

answer: let ~F (x, y, z) = ((x + 1)2, 0,x2) and C is the intersection of the cylinderx2 + 2x+ y2 = 3 and the plane z = x oriented counter clockwise as seen from above.

2. Compute the circulationC~F d~x directly and then use Stokes theorem to verify

your answer: let ~F (x, y, z) = (2y, z,z) and C is the intersection of the cylinderx2 + z2 = 1 and the plane y = x+ 1 oriented counter clockwise as seen from the origin.

3. Let ~F be a constant vector field. A surface in R3 and its boundary curve ~g areassumed to satisfy the assumptions of Stokes Theorem. Show that

~F n d = 12

~g

(~F ~x) d~x,

where ~x = (x, y, z). Hint: start with the right hand side.

4. Show that if the boundary curve and the surface satisfy the assumptions ofStokes theorem then

a)

(~f ~g

) n d =

f ~g d~x.

b) ~g

f ~f d~x = 0.

5. Consider Maxwells equations with no density charge and no current.

a) Take the curl of Faradays law and obtain (for some constant c)

2 ~E

t2= c22 ~E.

b) Take the curl of Ampe`res law and obtain (for some constant c)

2 ~B

t2= c22 ~B.

c) The constant c is the speed at which the solution propagates. What is c?

Note the identity ~ ~ ~F = ~(~ ~F

)2 ~F .

6. In a Perfect Fluid the pressure, p(~x, t) experts a force per unit area on a surface givenby pij nj times the surface element. Note that this is the force in the ei directiondue to the surface in the nj direction. Gravity exerts a force per unit mass on the fluidgiven by the constant vector ~g (gravitational acceleration). Furthermore, we define(~x, t) to be the density per unit volume and ~u(~x, t) to be the velocity.

a) Newtons second law states that the time derivative of the total momentum isequal to the sum of the forces. Use this principle to derive the following identity,

d

dt

V

~u dV =

V

~g dV

pijnjd.

where V (t) is the volume and (t) is the surface boundary.

b) Use Gauss Divergence Theorem to rewrite the pressure term as a triple integralin the equation from part a).

c) In the case where the volume is moving one needs to use a more general theoremcalled the Reynolds Transport theorem, which states,

d

dt

V

f dV =

V

f

t+ ~ (~uf) dV

Use this theorem and our lemma from class to obtain a differential equation thatdescribes the motion of a perfect fluid. This equation is called the Navier-Stokesequation and governs virtually all fluid motions.