Multiple Integrals, Vector Fields, Hemispheres and Divergence Theorem

Three Hours
UNIVERSITY OF MANCHESTER

Multiple Integrals, Vector Field Theory and Tensors MT2121

Tuesday 15th January, 2002
9.45a.m. ­ 12.45p.m.


The use of electronic calculators is NOT permitted.




Answer ALL seven questions in SECTION A (39 marks in all)
and
THREE of the four questions in SECTION B (12 marks each)
The total number of marks on this paper is 75. A further 25 marks are available from
the coursework during the semester, making a total of 100.
This module is worth 20 credits, so marks gained will be counted twice.


You may use the following formulae in Sections A and B without proof:-
r ^
In cylindrical polar coordinates (r, , z), where F = Fr ^ + F + Fz z,
^

1 1 F Fz
(1) div F = (rFr ) + +
r r r z
and
1 Fz F Fr Fz ^ 1 Fr
(2) curl F = - ^+
r - + (rF ) - ^
z.
r z z r r r

^ ^ ^
In spherical polar coordinates (R, , ), where F = FR R + F + F ,

sin
(3) (R sin ) div F = (R2 FR ) + (sin F ) + F
R R
and
^
(4) (R sin ) curl F = (sin F ) - F R

^ ^
+ FR - (R sin F ) + (RF ) - FR sin .
R R




P.T.O.
2
MT2121 January 2002 continued. . .


SECTION A
Answer ALL seven questions




A1. For each of the following equations, state whether it is valid or not (in terms of
the suffices only). If an equation is not valid, state the reason and one way to correct
it.
(i) dij bjk = ip jq dpq jq kr bqr ;
(ii) Clmn = Dluv Knvm Xu ;
(iii) Pzy = Qzx Rxy + Sz Tyu .
[5 marks]



A2. Verify the formula
zmn zst = ms nt - mt ns
in the case m = 1 and n = 2. Do this by finding the value of the left-hand and right-hand
sides in each of the four cases (i) s = 3, (ii) t = 3, (iii) s = t = 3 and (iv) all other possible
values of s and t. [5 marks]



A3. With the aid of a clear sketch of the region of integration, reverse the order of
integration in the integral
1 x x2
I= dy dx
0 0 1 + y3
to show that I = 1/4.
[You may use the formula (1 - y 6 ) = (1 - y 3 )(1 + y 3 ). ] [5 marks]



A4. The surfaces S1 and S2 are respectively the sphere x2 + y 2 + (z - 2a)2 = 2a2 and
the cone z 2 = x2 + y 2 (where a is a constant). You are given that the sphere sits inside
and touches the cone; show that the circle where they touch lies in the plane z = a and
has radius a.
Find the volume trapped between S1 and S2 .
[Your solution should include a clear sketch of the region under consideration.]
[6 marks]




MT2121 continued/3. . .
3
MT2121 January 2002 continued. . .

A5. Write down the formula for grad P , where P = P (R, , ) is given in spherical polar
coordinates, and use the given formula (3) for the divergence to write down 2 P .
You are given that P has the form

P (R, , ) = f (R) + ln(sin )

for some function f (R); show that

2 1 d df
P = R2 -1 .
R2 dR dR

You are given also that F = grad P is solenoidal, and that f (R) = f (R) = 0 on R = 1.
Find the function f (R). [6 marks]



A6. The vector field F is given by

F = (y 2 + 2y - x2 )i + 2x(y + 1)j + 7xk

(where is a constant). The loop C in the (x, y) plane consists of the following three
parts: (i) the parabola y = x2 from the origin to the point (1, 1, 0), (ii) the straight line
y = 1 to the point (0, 1, 0) and (iii) the straight line x = 0 back to the origin.
By direct calculation of the contributions from the three parts of C, evaluate the loop
integral c F · dr.
From your results, say whether you can deduce that F is, or is not, a conservative
field in the two cases = ±1. [7 marks]



A7. Write down the Cartesian components of curl A, and use these to show that

div (curl A) 0

for any vector field A with sufficiently smooth components.
Assume that, for some suitably smooth functions (x, y) and (x, y), A = (x, y)k
and curl A = grad . Write down the Cartesian components of the last equation, and
show that each of and satisfies Laplace's equation, that is 2 = 0 and 2 = 0.
[5 marks]




MT2121 P.T.O.
4
MT2121 January 2002 continued. . .


SECTION B
Answer THREE of the four questions




B8. (a) You are given that the rotation matrix A, from the unprimed frame to another
frame, has components pq which satisfy pq pr = qr = qp rp .
You are also given that B and C are both second-order tensors and have components
bij and cij in the unprimed frame.
Define in component form the tensor product D = BC and show that D is a fourth-
order tensor. Show also that the quantity E, which has components eik = bij cjk in the
unprimed frame, is another second-order tensor. [6 marks]
(b) The primed frame is now taken to be the one obtained from the unprimed frame

by a rotation about the 0x1 axis (so that 0x3 coincides with 0x2 ). Sketch the two sets
2
of axes, and find all the components ij of the rotation matrix A; here ij is the cosine
of the angle between 0xi and 0xj .
Show that A is orthogonal, state the transformation law applicable to B, and use this
to find the four components b12 , b13 , b22 and b23 of B in the primed frame. [6 marks]




MT2121 continued/5. . .
5
MT2121 January 2002 continued. . .

B9. (a) Show that the Jacobian of the transformation

x2 y2
u= and v =
y x
is a constant. Use this transformation to show that
14
(3y 3 - 2x3 )dx dy = ,
R 9

where R is the region bounded between the parabolas y = x and y = 1 x2 and the 3
straight line y = x. [Your solution should include a sketch of the region of integration R,
and its image under the transformation.] [6 marks]
(b) The surface S is that part of the spheroid
1
x2 + y 2 + z 2 = 3a2
4
which lies inside the paraboloid az = x2 + y 2 ; here a is a constant. Sketch the surface S
and the paraboloid by drawing their intersections with the plane y = 0.
Show that
z dS = 2 3 a2 + x2 + y 2 dx dy,
S R

where R is a region of the (x, y) plane you should find, and hence evaluate the surface
integral (using plane polar coordinates in the plane integral). [6 marks]




MT2121 P.T.O.
6
MT2121 January 2002 continued. . .

B10. (a) State the Divergence Theorem, being careful to explain any notation you use
and any conditions that must apply.
The vector field B is given by
^ ^
B = R cos (cos R - sin )

in spherical polar coordinates (R, , ). This field exists in a region which includes the
hemisphere x2 + y 2 + z 2 a2 , z 0. By direct evaluation of the two surface integrals,
find the flux B · n dS of B out of the (closed) surface of the hemisphere. [Hint: this
^
surface consists of two parts, a hemispherical cap and a flat base, which are respectively
R = const and = const.]
Now use the given formula (3) for the divergence to show that div B = 1, and confirm
your result for the flux by an application of the Divergence Theorem. [8 marks]
(b) The vector field F is given by

F = (x2 y 2 + 2xe-y )i + (2x3 y - x2 e-y + 1)j

where is a constant. For a certain value of , which you should find, the line integral
r
S(r) = F · dr
0

is independent of the path from 0 to r. Find S(r) for this particular value of .
[4 marks]



B11. State Stokes' Theorem, being careful to explain any notation you use and paying
particular attention to the direction of all vectors.
In spherical polar coordinates (R, , ) the vector field B is given by
^ ^
B = R2 cos3 R - R2 sin (cos2 + 1) .

By direct calculation, find the flux B · n dS of B through the part of the sphere
^
2 2 2 2
x + y + z = a with z h (where a > h > 0). [The substitution c = cos may be
useful.]
Now use the given formula (4) to show that B = curl A, where
1 ^
A = R3 sin (cos2 + 1) ,
4
and use Stokes' Theorem to confirm your result for the flux. [12 marks]




MT2121