UNIVERSITY OF DUBLIN
XMA1132
TRINITY COLLEGE
Faculty of Engineering, Mathematics
and Science
school of mathematics
JF Mathematics
JF Theoretical Physics
JF TSM Mathematics
Hilary Term 2014
Module MA1132,
???, ???
Final Exam
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9.30 — 11.30
Dr. Sergey Frolov
ATTEMPT FOUR QUESTIONS
Log tables are available from the invigilators, if required.
Non-programmable calculators are permitted for this examination,—please indicate the make
and model of your calculator on each answer book used.
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Each question is worth 20 marks. Show the details of your work.
Sample I
1. The equations of motion of a system of n particles are given by
mi ẍi = −
∂U (x1 , . . . , xn )
,
∂xi
ẍi =
d 2 xi
,
dt2
i = 1, 2, . . . , n ,
where mi is the mass and xi is the coordinate of the i-th particle, and U (x1 , . . . , xn ) is
the potential energy of the system.
Consider a system of n coupled anharmonic oscillators with the potential
U (x1 , . . . , xn ) =
n
X
λ
(xi+1 − xi ) +
(xi − xj )4 .
2
4
i,j=1
n−1
X
κ
i=1
2
(a) 3 marks. Find the equations of motion of the first particle (x1 ).
(b) 3 marks. Find the equations of motion of the second particle (x2 ).
(c) 3 marks. Find the equations of motion of the last particle (xn ).
(d) 8 marks. Find the equations of motion of the i-th particle (xi ) for 1 < i < n.
(e) 3 marks. Write the equations of motion of the i-th particle (xi ) for 1 ≤ i ≤ n by
using the Kronecker delta δij .
Answer:
mi ẍi = −κ(2xi − xi−1 − xi+1 ) + κx1 δi1 + κxn δin − 2
n
X
λ(xi − xj )3 ,
(1)
j=1
where we assume that x0 = xn+1 = 0.
2. Consider the function
f (x, y, z) =
p
2xy + 3z 4 − 6 cos(3x − 2y) ,
and the point P = (2, 3, −1) .
(a) 12 marks. Find a unit vector in the direction in which f increases most rapidly at
the point P .
(b) 1 mark. Sketch the projection of this vector onto the xz-plane
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(c) 3 marks. Find a unit vector in the direction in which f decreases most rapidly at
the point P .
(d) 1 mark. Sketch the projection of this vector onto the xy-plane
(e) 3 marks. Find the rate of change of f at the point P in these directions.
Answer:
(a) u =
3
7
1 , 32 , −2 ≈ (0.428571, 0.285714, −0.857143) .
(b) uxz = 73 (1 , 0 , −2) ≈ (0.428571 , 0 , −0.857143) .
(c) v = − 73 1 , 23 , −2 ≈ (−0.428571, −0.285714, 0.857143) .
(d) vxy = − 37 1 , 23 , 0 ≈ (−0.428571, −0.285714, 0) .
(e)
7
3
≈ 2.33333 and − 37 ≈ −2.33333 .
3. Consider the intersection of the surfaces
z = λ x + µ y + h , λ > 0 , µ > 0 , h > 0 , and
y2
x2
+
= 1.
a2
b2
(a) 1 mark. What is the surface z = λ x + µ y + h ?
(b) 1 mark. What is the surface
x2
a2
+
y2
b2
=1?
(c) 2 mark. Sketch the surfaces for a = b = λ = µ = h = 1.
(d) 12 mark. Find the coordinates of the point on the intersection which has the
maximum z-coordinate.
(e) 4 mark. Find the coordinates of the point on the intersection which has the
minimum z-coordinate.
Answer: See, PS7, Q5d.
4. Consider the sum of the integrals
Z 1Z y
Z
f (x, y)dxdy +
0
y 2 /9
1
3
Z
1
f (x, y)dxdy .
y 2 /9
(a) 6 marks. Determine and sketch the integration region R.
(b) 3 marks. Write the sum as one repeated integral by reversing the order of integration.
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(c) 11 marks. Compute the integral if
y ex
.
9−x
f (x, y) =
Answer: See Tutorial7, Q3.
Z 1Z
0
y
y 2 /9
y ex
dxdy +
9−x
Z
1
3
Z
1
y 2 /9
y ex
1
dxdy = .
9−x
2
5. (a) 1 marks. Express rectangular coordinates in terms of spherical coordinates, and
draw the corresponding picture.
(b) Consider the solid G bounded above by the surface x2 + y 2 + z 2 = 16 and below
p
by the surface z = x2 + y 2 .
i. 1 marks. What is the surface x2 + y 2 + z 2 = 16?
p
ii. 1 marks. What is the surface z = x2 + y 2 ?
iii. 2 marks. Sketch the solid G.
iv. 5 marks. Use a triple integral and spherical coordinates to compute the volume
V of the solid G.
v. 7 marks. Use a triple integral and spherical coordinates to find the mass M
of the solid G if its density is
2
2
2
1 − e−(x +y +z )
δ(x, y, z) = p
.
x2 + y 2 + z 2
vi. 3 marks. What is the density of the solid G at the origin: δ(0, 0, 0) =?
Answer:
64 √
2 − 2 π ≈ 39.2598 ,
V =−
3√
2 − 2 (1 + 15e16 ) π
M =−
≈ 13.8023 ,
2e16
δ(0, 0, 0) = 0 .
(2)
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Sample II
1. The equations of motion of a system of n particles are given by
mi ẍi = −
∂U (x1 , . . . , xn )
,
∂xi
ẍi =
d 2 xi
,
dt2
i = 1, 2, . . . , n ,
where mi is the mass and xi is the coordinate of the i-th particle, and U (x1 , . . . , xn ) is
the potential energy of the system.
Consider a system of n particles with the potential
U (x1 , . . . , xn ) =
n−1
X
i=1
n
X
λ 4
xi .
cosh κ(xi+1 − xi ) +
4
i=1
(a) 3 marks. Find the equations of motion of the first particle (x1 ).
(b) 3 marks. Find the equations of motion of the second particle (x2 ).
(c) 3 marks. Find the equations of motion of the last particle (xn ).
(d) 8 marks. Find the equations of motion of the i-th particle (xi ) for 1 < i < n.
(e) 3 marks. Write the equations of motion of the i-th particle (xi ) for 1 ≤ i ≤ n by
using the Kronecker delta δij .
Answer:
mi ẍi = −κ(sinh κ(xi − xi−1 ) + sinh κ(xi − xi+1 )) + κ sinh κx1 δi1 + κ sinh κxn δin − λx3i ,
(3)
where we assume that x0 = xn+1 = 0.
2. Consider the surface
r
z = f (x, y) = ln
3
12 sin(x − 2y) + 8y 2 − x3 − 6x2 y + 32
8e2
!
.
(a) 9 marks. Find an equation for the tangent plane to the surface at the point
P = (2, 1, z0 ) where z0 = f (2, 1).
(b) 3 marks. Find points of intersection of the tangent plane with the x-, y- and
z-axes.
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(c) 2 marks. Sketch the tangent plane, and show the point P = (2, 1, z0 ) on it.
(d) 4 marks. Find parametric equations for the normal line to the surface at the point
P = (2, 1, z0 ).
(e) 2 marks. Sketch the normal line to the surface at the point P = (2, 1, z0 ).
Answer: See PS5, Q5.
3. Consider a triangle which has the sides a, b, c and the angle between the sides a and b
is φ, see the picture.
(a) 1 mark. What is the perimeter of the triangle?
(b) 1 mark. What is the area of the triangle?
(c) 5 marks. What is the constraint which relates a, b, c and φ?
(d) 13 marks. Show that a triangle with fixed perimeter has maximum area if it is
equilateral.
Answer: See PS6, Q5.
4. Consider the integral
√
Z
1
−7
Z
2+
√
2−
7−6y−y 2
f (x, y)dxdy .
7−6y−y 2
(a) 6 marks. Determine and sketch the integration region R.
(b) 3 marks. Reverse the order of integration.
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(c) 11 marks. Compute the integral if
f (x, y) = x y .
Answer: See PS7, Q2(a).
√
Z
1
−7
Z
2+
√
2−
7−6y−y 2
x y dxdy = −96π .
7−6y−y 2
5. (a) 1 mark. Express rectangular coordinates in terms of cylindrical coordinates
(b) Consider the solid G bounded above by the surface z = 1/rα , α > 0, r =
p
x2 + y 2 and below by the plane z = 1. The surface z = 1/r is Gabriel’s Horn.
i. 2 marks. Sketch the solid G for α = 1/3 , α = 1, and its projection onto the
xy-plane.
ii. 4 marks. Find the volume V of the solid G, and determine for which values
of α it is finite.
iii. 5 marks. Find the centroid of the solid G, and determine for which values of
α it is finite.
iv. 8 marks. Let the density function δ(x, y, z) of the solid G be equal to 1. Then
the moments of inertia Ix , Iy , and Iz about the x-axis, the y-axis, and the
z-axis, respectively, are given by
ZZZ
ZZZ
2
2
Ix =
(y +z )dV , Iy =
(x2 +z 2 )dV ,
G
G
ZZZ
(x2 +y 2 )dV .
Iz =
G
Find the moments of inertia of the solid G, and determine for which values of
α they are finite.
Answer: See PS9, Q3.
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Sample III
1. The equations of motion of a system of n particles are given by
mi ẍi = −
∂U (x1 , . . . , xn )
,
∂xi
d 2 xi
,
dt2
ẍi =
i = 1, 2, . . . , n ,
where mi is the mass and xi is the coordinate of the i-th particle, and U (x1 , . . . , xn ) is
the potential energy of the system.
Consider a system of n particles with the Toda potential
U (x1 , . . . , xn ) =
n−1
X
eα(xi+1 −xi ) .
i=1
(a) 3 marks. Find the equations of motion of the first particle (x1 ).
(b) 3 marks. Find the equations of motion of the second particle (x2 ).
(c) 3 marks. Find the equations of motion of the last particle (xn ).
(d) 8 marks. Find the equations of motion of the i-th particle (xi ) for 1 < i < n.
(e) 3 marks. Write the equations of motion of the i-th particle (xi ) for 1 ≤ i ≤ n by
using the Kronecker delta δij .
Answer:
mi ẍi = −αeα(xi −xi−1 ) + αeα(xi+1 −xi ) + αeα x1 δi1 − αe−α xn δin ,
where we assume that x0 = xn+1 = 0.
2. Let z = f (x, y) = f˜(r, θ), x = r cos θ, y = r sin θ.
(a) 8 marks. Express
∂z
∂z
and
∂x
∂y
in terms of
∂z ∂z
,
, r and θ .
∂r ∂θ
(b) 10 marks. Express
∂ 2z
∂ 2z
and
∂x2
∂y 2
in terms of
∂ 2 z ∂ 2 z ∂ 2 z ∂z ∂z
,
,
,
,
, r and θ .
∂r2 ∂θ2 ∂r∂θ ∂r ∂θ
(c) 2 marks. Show that
∂ 2z ∂ 2z
∂ 2 z 1 ∂z
1 ∂ 2z
+
=
+
+
.
∂x2 ∂y 2
∂r2 r ∂r r2 ∂θ2
Answer: See PS5, Q4.
(4)
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3. Consider the intersection of the surfaces
z=
p
c2 − x2 − y 2 , and
(a) 1 mark. What is the surface z =
(b) 1 mark. What is the surface
x2
a2
x2
y2
+
= 1,
a2
b2
c ≥ a ≥ b.
p
c2 − x 2 − y 2 ?
+
y2
b2
=1?
(c) 2 mark. Sketch the surfaces for c = 3, a = 2, b = 1.
(d) 12 mark. Use Lagrange multipliers to find the coordinates of the points on the
intersection which have the maximum z-coordinate.
(e) 4 mark. Use Lagrange multipliers to find the coordinates of the points on the
intersection which have the minimum z-coordinate.
Answer:
P (zmax ) = (0, ±b,
√
c2 − b2 ) ,
√
P (zmin ) = (±a, 0, c2 − a2 )
(5)
4. Consider the solid G bounded by the surfaces y = 1, z = 0, y = x2 , and z = x2 + y 2 .
(a) 1 mark. What are the surfaces y = 1 and z = 0?
(b) 1 mark. What is the surface y = x2 ?
(c) 1 mark. What is the surface z = x2 + y 2 ?
(d) 2 marks. Sketch the solid G, and its projection onto the xy-plane.
(e) 6 marks. Compute the volume V of the solid G.
(f) 9 marks. Find the centroid of the solid G.
Answer:
V =
88
≈ 0.838095 ,
105
(xcg , ycg , zcg ) = (0,
25 1006
,
) ≈ (0, 0.757576, 0.923783) .
33 1089
(6)
5. (a) 1 mark. Express rectangular coordinates in terms of cylindrical coordinates
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(b) Consider the solid G bounded above by the surface
15 p 2
1
− x + y 2 and below by the surface z = (x2 + y 2 ) .
2
2
p
i. 1 mark. What is the surface z = 15
− x2 + y 2 ?
2
z=
ii. 1 mark. What is the surface z = 12 (x2 + y 2 )?
iii. 2 marks. Sketch the solid G, and its projection onto the xy-plane.
iv. 5 marks. Use a triple integral and cylindrical coordinates to compute the
volume V of the solid G.
v. 7 marks. Use a triple integral and cylindrical coordinates to find the mass M
of the solid G if its density is
√
2
2
e x +y − 1
p
δ(x, y, z) =
.
x2 + y 2 + 5 x2 + y 2
vi. 3 marks. What is the density of the solid G at the origin: δ(0, 0, 0) =?
Answer: See Tutorial9, Q2.
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Useful Formulae
1. Let r(t) be a vector function with values in R3 :
(a) The unit normal vector is N(t) =
dT
dt
dT
| dt |
r(t) = x(t) i + y(t) j + z(t) k .
.
(b) The unit binormal vector is B(t) = T(t) × N(t) .
(c) The curvature of C is κ(t) =
|T0 (t)|
.
|r0 (t)|
2. Let σ be a surface in R3 : z = f (x, y)
(a) The slope kx of the surface in the x-direction at (x0 , y0 ) is kx =
∂z
(x0 , y0 ) .
∂x
(b) The slope ky of the surface in the y-direction at (x0 , y0 ) is ky =
∂z
(x0 , y0 ) .
∂y
(c) The equation for the tangent plane to the surface at the point P = (x0 , y0 , z0 ) is
z = z0 + kx (x − x0 ) + ky (y − y0 ) .
(d) Parametric equations for the normal line to the surface at P = (x0 , y0 , z0 ) are
r(t) = r0 + t(−kx i − ky j + k) , r0 = x0 i + y0 j + z0 k .
(e) The mass of the lamina with the density δ(x, y, z) that is the portion of the surface
that is above a region R in the xy-planeris
RR
RR
M = σ δ(x, y, z) dS = R δ(x, y, z) 1 +
∂z 2
∂x
+
2
∂z
∂y
dA .
3. Let R be a plain lamina with density δ(x, y).
(a) Its mass is equal to M =
RR
R
δ(x, y) dA .
(b) The x-coordinate of its centre of gravity is equal to xcg =
1
M
RR
(c) The y-coordinate of its centre of gravity is equal to ycg =
1
M
RR
R
x δ(x, y) dA .
R
y δ(x, y) dA .
4. The volume element in spherical coordinates is dV = r2 sin φ dr dφ dθ .
5. Let a region Rxy in the xy-plane be mapped to a region Ruv in the uv-plane under the
change of variables u = u(x, y) , v = v(x, y).
∂u ∂v ∂u ∂v (a) The magnitude of the Jacobian of the change is ∂(u,v)
=
−
.
∂(x,y)
∂x ∂y
∂y ∂x −1
RR
RR
(b) The integral over Rxy is Rxy f (x, y) dAxy = Ruv f (x(u, v), y(u, v)) ∂(u,v)
dAuv .
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