A. G. Walton
M2AA2 Multivariable Calculus
1
Problem Sheet 1
1. If A is a constant vector field, calculate the gradients of the following scalar fields:
(i)
(ii)
(iii)
A ∙ r,
rn ,
r ∙ ∇(x + y + z),
where r = xi + yj + zk and r = |r|.
2. If φ = x2 y + z 2 x and P is the point (1, 1, 2), find the directional derivative of φ at P in the direction
(1, 2, 3).
3. Find the equations of the tangent planes to the following surfaces at the points indicated
(i)
(ii)
x2 + 2y 2 − z 2 − 8 = 0 at (1, 2, 1),
z = 3x2 y sin(πx/2) at x = 1, y = 1.
4. If φ = xr 2 , r = xi + yj + zk and f (r) is an arbitrary function of r = |r| , evaluate:
(i)
(ii)
(iii)
∇φ,
div (φr),
curl (f (r)r).
2
5. If u = z 2 i, v = xi + yj + zk and ψ = |v| , verify the identities
(i)
(ii)
6. Use tensor notation and
identities:
(i)
(ii)
(iii)
div (u × v) = v ∙ curl u − u ∙ curl v,
div (ψu) = (∇ψ) ∙ u + ψdiv u.
the relation εijk εilm = δjl δkm − δjm δkl to establish the following vector
(a × b) ∙ (a × b) = (a ∙ a)(b ∙ b) − (a ∙ b)2 ,
(a × b) ∙ (c × d) = (a ∙ c)(b ∙ d) − (b ∙ c)(a ∙ d),
(a × b) × (c × d) = ((a × b) ∙ d))c − ((a × b) ∙ c)d,
7. Simplify the following expressions:
(i)
(ii)
(iii)
(iv)
(v)
δij ∂xi /∂xj ,
δij δik xj xk ,
δij ∂ 2 φ/∂xi ∂xj ,
δij δjk δki ,
εijk ∂/∂xi (∂Ak /∂xj ) .
8. Use tensor notation to prove the following identities:
(i)
(ii)
(iii)
curl (φA) = φ curlA + ∇φ × A,
div (A × B) = B ∙ curl A − A ∙ curl B,
2
A × curl A = 12 ∇(|A| ) − (A ∙ ∇)A.
A. G. Walton
M2AA2 Multivariable Calculus
Sheet 1 Answers
1.
2.
3.
4.
7.
n−2
(i) A;
r; (iii) i + j + k.
√ (ii) nr
20/ 14.
(i) x + 4y − z = 8; (ii) 6x + 3y − z = 6.
(i) (3x2 + y 2 + z 2 )i + 2xyj + 2zxk; (ii) 6xr 2 ; (iii) zero.
(i) 3; (ii) r2 ; (iii) ∂ 2 φ/∂x21 + ∂ 2 φ/∂x22 + ∂ 2 φ/∂x23 ; (iv) 3; (v) zero.
2
A. G. Walton
M2AA2 Multivariable Calculus
1
Problem Sheet 2
1. The vector field v is given by
v = (2xy + z 2 ) i + (2yz + x2 ) j + (2xz + y 2 ) k.
Show that curl v = 0 and find the potential φ such that v = ∇φ with φ = 0 at the origin. Hence evaluate
the line integral
Z
P
v ∙ dr,
where P is any path joining (0, 0, 0) to (1, 2, 3).
2. Evaluate the line integral
I=
Z
(xy dx + yz dy + zx dz)
P
where P is the straight line joining the starting point A(0, 0, 0) and the end point B(1, 2, 3).
3. The vector field F is given by
Evaluate the line integral
F = 3x2 i + (2xz − y) j + z k.
Z
P
F ∙ dr
along each of the following paths between (0, 0, 0) and (2, 1, 3) :
(i)
(ii)
(iii)
a straight line;
the curve defined by (2t2 , t, 4t2 − t) with 0 ≤ t ≤ 1;
the curve defined by (s, s2 /4, 3s3 /8) with 0 ≤ s ≤ 2.
4. In each of the following cases (a) sketch the region of integration; (b) evaluate the integral; (c) write
down the integral with the order of integration reversed; (d) evaluate the integral again (if possible) and
compare with (b):
R a R a−x Ra Rx 2
(i)
dy dx;
(ii)
(x + y 2 ) dy dx;
0 0
0
0
R 1 R √x
R 1 R x
2
(iii) 0 x xy 2 dy dx (iv) 0 0 e−x dy dx.
5. If
F = 2y i − z j + x2 k,
and S is the surface of the parabolic cylinder y 2 = 8x in the first octant, bounded by the planes y = 4
and z = 6, evaluate
Z
b dS,
F∙n
S
b points in the direction of increasing x, by projecting the integral onto the plane x = 0.
where n
6. If
F = y i + (x − 2xz) j − xy k
and S is the surface of the sphere x2 + y 2 + z 2 = a2 above the x − y plane, evaluate
Z
b dS,
(∇ × F) ∙ n
S
b is the unit normal out of the sphere. (Hint: project onto the x − y plane and use plane polar
where n
coordinates (r, θ) to evaluate the resulting integral - you may assume that dxdy = rdrdθ).
A. G. Walton
M2AA2 Multivariable Calculus
2
7. Verify Green’s theorem in the plane
Z I
∂M
∂L
(L dx + M dy) =
dx dy
−
∂x
∂y
C
R
for the special case where C is the boundary of the rectangle with vertices (0, 0), (a, 0), (a, b), (0, b) and
L = ay, M = 2xy.
8. Use Green’s theorem to show that the area enclosed by a simple closed curve with boundary C can
be expressed as
I
1
(x dy − y dx).
2 C
Use this result to calculate the area bounded by one arc of the cycloid
x = a(t − sin t), y = a(1 − cos t),
(with a > 0, and 0 ≤ t ≤ 2π) and the x−axis.
Sheet 2 Answers
1.
2.
3.
4.
5.
6.
7.
8.
φ = yx2 + xz 2 + zy 2 ; value of integral is 23.
23/3.
(i) 16; (ii) 71/5; (iii) 16.
(i) a2 /2; (ii) a4 /3; (iii) 1/35; (iv) 12 (1 − e−1 ).
132.
zero.
LHS = RHS = ab2 − a2 b.
3πa2 .
A. G. Walton
M2AA2 Multivariable Calculus
1
Problem Sheet 3
1. A region V is enclosed by a surface S. In V the vector field is solenoidal. If φ is a scalar field that
takes a constant value on the surface S, show that
Z
A ∙ ∇φ dV = 0.
V
(Hint: consider the divergence theorem applied to φ A).
2. Evaluate
Z
S
b dS
r∙n
where S is any closed surface enclosing a volume V, and r is the position vector xi + yj + zk.
3. Show that
Z
S
b
r∙n
dS =
2
r
Z
V
dV
,
r2
where S is a closed surface enclosing the volume V, and r is as above, with r = |r| .
4. Use the divergence theorem to prove the following results, where S is a closed surface with unit outward
b enclosing the volume τ , φ(x, y, z) is a scalar and A(x, y, z) a vector function of position.
normal n
Z
Z
Z
Z
b φ dS =
b × A dS =
n
n
(i)
∇φ dτ. (ii)
curl A dτ.
S
τ
S
τ
5. Verify the divergence theorem for the case when
A = xi
and V is the cube |x| ≤ a, |y| ≤ a, |z| ≤ a.
6. Let S be the piecewise smooth closed surface consisting of the surface of the cone z = (x2 + y 2 )1/2 for
x2 + y 2 ≤ 1, together with the flat cap consisting of the disk x2 + y 2 ≤ 1 in the plane z = 1. Verify the
divergence theorem
Z
Z
S
b dS =
A∙n
div A dV
V
for this surface, when A = (x + y)i + (y − x − z)j + (z − y)k. You may assume dx dy = r dr dθ in plane
polar coordinates.
7. By converting into an appropriate line integral, use Stokes theorem to evaluate
Z
b dS
(∇ × A) ∙ n
S
where A = (y − z, z − x, x − y). Here S is the upper half of the ellipsoid x2 /a2 + y 2 /b2 + z 2 /c2 = 1 with
b is the unit outward normal to S.
z ≥ 0, and n
8. Verify Stokes theorem for the vector field
A = (3x − y, −yz 2 /2, −y 2 z/2)
where S is the upper half surface of the sphere x2 + y 2 + z 2 = a2 , so that the closed curve C is a circle
in the x − y plane.
Hint: to evaluate the surface integral use spherical polar coordinates x = a sin θ cos φ, y = a sin θ sin φ, z =
a cos θ, with dS = a2 sin θ dθ dφ, and 0 ≤ θ ≤ π/2, 0 ≤ φ ≤ 2π.
A. G. Walton
M2AA2 Multivariable Calculus
9. Let S consist of the part of the cone z = (x2 + y 2 )1/2 for x2 + y 2 ≤ 9 and suppose
A = (−y, x, −xyz).
Verify that Stokes theorem is satisfied for this choice of A and S.
Sheet 3 Answers
2.
5.
6.
7.
8.
9.
3V .
LHS = RHS = 8a3 .
LHS = RHS = π.
−2πab.
LHS = RHS = πa2 .
LHS = RHS = 18π.
2
A. G. Walton
M2AA2 Multivariable Calculus
1
Problem Sheet 4
1. Bipolar coordinates (ξ, η, z) are given in terms of Cartesian coordinates (x, y, z) by
x=
c sinh ξ
c sin η
, y=
, z = z,
cosh ξ − cos η
cosh ξ − cos η
where c is a constant. Show that this coordinate system is orthogonal, and find the scale factors h1 , h2 , h3 .
2. Consider the following curvilinear coordinate system (u, v, z) defined in terms of the Cartesian coordinates (x, y, z) as:
1
x = (u2 − v 2 ), y = uv, z = z,
2
e1 , b
e2 , b
e3 of the curvilinear
with u ≥ 0. Find the scale factors h1 , h2 , h3 and express the unit vectors b
coordinate system in terms of the Cartesian unit vectors i, j, k.
3. The vector field F is given in terms of the curvilinear coordinate system described in Q2 as
e1 − v(u2 + v 2 )3/2 b
e2 .
F = u(u2 + v 2 )3/2 b
Show that
div F = 4(u2 − v 2 ),
curl F = −8uv b
e3 ,
F = 4(x2 + y 2 ) i.
(i)
(ii)
(iii)
Using the result in (iii), confirm your answers to (i) and (ii) by carrying out the calculations in Cartesian
coordinates.
bb
4. Find the unit vectors b
r, φ,
z of a cylindrical polar coordinate system (r, φ, z) in terms of i, j, k. Solve
bb
for i, j, k in terms of b
r, φ,
z. Represent the vector
F = y i + z j + xk
in cylindrical coordinates and determine Fr , Fφ and Fz .
5. Using the substitution
u = y 2 − x2 , v = 2xy,
evaluate the integral
Z
(x2 + y 2 )3 dx dy
R
where R is the finite region in the first quadrant, bounded by the curves x2 − y 2 = 1, y 2 − x2 = 1, xy = 1
and xy = 2.
6. Use plane polar coordinates to evaluate the integral
Z
(x4 + y 4 ) dx dy
R
2
2
where R is the circular disc x + y ≤ 1.
7. Use the transformation
u = x − y, v = x + y,
to evaluate
Z
R
(x + y)2 cos(x2 − y 2 ) dx dy
where R is the region in the x − y plane enclosed by the lines y = 0, x = 0 and y = 1 − x.
A. G. Walton
M2AA2 Multivariable Calculus
8. Find the surface area of the parameterized helicoid
x = λ cos s, y = λ sin s, z = s,
with 0 6 λ 6 1 and 0 6 s 6 2π.
9. Find
Z
z 2 dS,
S
where S is the surface of a torus, parameterized as
x = (a + b cos t) cos θ, y = (a + b cos t) sin θ, z = b sin t,
with 0 6 θ 6 2π, 0 6 t 6 2π and a > b > 0.
Sheet 4 Answers
1. h1 = c/(cosh ξ − cos η) = h2 ; h3 = 1.
e1 = (ui + vj)/(u2 + v 2 )1/2 , b
e2 = (−vi + uj)/(u2 + v 2 )1/2 , b
e3 = k.
2. h1 = (u2 + v 2 )1/2 = h2 ; h3 = 1; b
b
4. b
r = i cos φ + j sin φ, φ = −i sin φ + j cos φ, b
z = k;
i=b
r cos φ − φb sin φ, j = b
r sin φ + φb cos φ;
Fr = r sin φ cos φ + z sin φ, Fφ = −r sin2 φ + z cos φ, Fz = r cos φ.
5. 29/3.
6. π/4.
7. 12 (1 − cos(1)).
√
8. π(sinh−1 (1) + 2).
9. 2π 2 ab3 .
2
A. G. Walton
M2AA2 Multivariable Calculus
1
Problem Sheet 5
1. Establish the following trigonometric identities:
(i)
N
X
n=1
sin θ cos[(2n − 1)θ] =
N
X
1
sin 2N θ; (ii) 1 +
2 cos nθ = sin[(N + 1/2)θ]/ sin(θ/2).
2
n=1
By integrating (i) with respect to x, deduce that
Z x
N
X
sin[(2n − 1)x]
sin 2N θ
=
dθ.
2n
−
1
2 sin θ
0
n=1
2. Find the Fourier series of period 2π which represent the following functions on the interval −π <
x<π:
(i) f (x) = x; (ii) f (x) = x2 ; (iii) f (x) = sinh x.
In each case sketch the function represented by the Fourier series in the range −3π < x < 3π, and find
the value to which the Fourier series converges at x = π.
3. Obtain the Fourier expansion
cos αx =
∞
sin απ X
2α sin απ
(−1)n
cos nx, |x| ≤ π,
+
απ
π(α2 − n2 )
n=1
when α is not an integer. What happens to the terms of the series when α → m, an integer?
4. Show that the Fourier series representation on (−π, π) of the function
1 + (x/π) , −π < x < 0,
f (x) =
1 − (x/π) , 0 ≤ x ≤ π,
is given by
f (x) =
∞
1
4 X cos ((2k + 1)x)
+ 2
.
2 π
(2k + 1)2
k=0
P∞
2
2
Deduce that k=0 1/(2k + 1) = π /8.
5. Find the Fourier series of period 2 that P
represents the function (x − 1)2 over the range 0 ≤ x ≤ 2.
∞
Using Parseval’s theorem deduce the value of n=1 1/n4 .
6. For the function f (x) = x(π − x), 0 ≤ x ≤ π, derive the Fourier half-range sine and cosine
expansions
∞
∞
π 2 X cos (2kx)
8 X sin ((2k − 1)x)
(i) f (x) =
,
(ii)
f
(x)
=
,
−
6
π
(2k − 1)3
k2
k=1
k=1
valid for 0 < x < π. Sketch the functions represented by the two series in the range −π < x < π. By
evaluating the series at appropriate points, find:
(a)
∞
∞
∞
X
X
X
1
(−1)n−1
(−1)n−1
,
(b)
,
(c)
.
2
2
n
(2n − 1)3
n
n=1
n=1
N =1
P∞
Use Parseval’s theorem to show that n=1 1/n6 = π 6 /945.
7. Show that the half-range Fourier sine series for f (x) = 1 + (x/L), 0 < x < L, is:
∞
nπx X
2
.
(1 − 2(−1)n ) sin
nπ
L
n=1
Sketch the function represented by the series in the range −2L < x < 2L.
A. G. Walton
M2AA2 Multivariable Calculus
Sheet 5 Answers
P∞
n+1
2. (i) x = n=1 (2(−1)
P∞ /n) sinn nx;2 FS = 0 at x = π.
2
2
(ii) x = π /3 + n=1 (4(−1)
nx; FS = π 2 at x = π.
P∞ /n ) cos
n+1
n/(1 + n2 ) sin nx; FS = 0 at x = π.
(iii) sinh x = (2/π) sinh π n=1 (−1)
3. As α → m all terms of the
series tend to zero,
that am → 1.
P∞
Pexcept
∞
4
4
5. (x − 1)2 = 1/3 + (4/π 2 ) n=1 cos(nπx)/n2 ;
n=1 1/n = π /90.
2
2
3
6. (a) π /6; (b) π /12; (c) π /32.
2
A. G. Walton
M2AA2 Multivariable Calculus
1
Problem Sheet 6
1. Find the Fourier transforms of the following functions: (with a > 0)
(i) f (x) = exp(−a |x|);
(ii) f (x) = sgn(x) exp(−a |x|); [sgn(x) = 1 if x > 0 and −1 if x < 0].
(iii) f (x) = 2a/(a2 + x2 ); (Hint: use the result of (i) and the symmetry formula from lectures)
(iv) f (x) = 1 − x2 for |x| ≤ 1 and zero otherwise;
(v) f (x) = sin(ax)/(πx); (Hint: use the transform of a rectangular pulse from the lectures and the
symmetry formula).
From your result in part (v), deduce that
Z ∞
0
sin x
π
dx = .
x
2
2. If a function has Fourier transform fb(ω), find the Fourier transform of f (x) sin(ax) in terms of fb.
3. By applying the inversion formula to the transforms obtained in 1(i) and 1(iv), establish the
following results:
Z ∞
Z ∞
cos x
sin x − x cos x
π
πe−a
(i)
dx
=
dx = .
if
a
>
0;
(ii)
2
2
3
x +a
x
2
2a
0
−∞
4. Sketch the function given by
f (x) =
2d − |x|
0
and show that fb(ω) = (2/ω)2 sin2 (ωd).
Use the energy theorem to demonstrate that
Z
∞
−∞
sin x
x
4
for |x| ≤ 2d,
,
otherwise.
dx =
2π
.
3
5. Show that the Fourier transform of exp(−cx)H(x),where H is the Heaviside function and c is a
positive constant, is given by 1/(c + iω). Hence use the convolution theorem to find the inverse Fourier
transform of
1
,
(a + iω)(b + iω)
where a > b > 0.
6. Use the symmetry rule to show that
F{f (x)g(x)} =
1 b
(f (ω) ∗ gb(ω)).
2π
7. Suppose that f (x) is a function such that fb(ω) = 0 for all ω with |ω| > M, where M is a positive
constant. Let g(x) = sin(ax)/(πx). Show that if the constant a > M :
f (x) ∗ g(x) = f (x).
Hint: Use the transform of g(x) from Q1(v).
8. By considering suitable integration formulae, establish the following results involving the Dirac
delta function:
(i) f (x)δ(x − x0 ) = f (x0 )δ(x − x0 ); (ii) xδ 0 (x) = −δ(x); (iii) δ(−x) = δ(x).
[In each case multiply by an arbitrary test function φ(x)and integrate from −∞ to ∞].
A. G. Walton
M2AA2 Multivariable Calculus
Sheet 6 Answers
1. (i) 2a/(a2 + ω 2 ); (ii) −2iω/(a2 + ω 2 ); (iii) 2π exp(−a |ω|); (iv) −(4/ω 2 ) cos ω + (4/ω 3 ) sin ω;
(v) 1 for |ω| ≤ a, zero otherwise.
2. (i/2)fb(ω + a) − (i/2)fb(ω − a).
5. (exp(−bx) − exp(−ax))/(a − b) if x > 0, zero if x < 0.
2
A. G. Walton
1
M2AA2 Multivariable Calculus
Problem Sheet 7
1. The wave equation (with a wavespeed of unity) is
∂2u
∂2u
=
, t ≥ 0, 0 ≤ x ≤ π.
∂t2
∂x2
Use the method of separation of variables to solve the equation with boundary conditions
u(0, t) = 0 and u(π, t) = 0 for all t > 0
and initial conditions
u(x, 0) = sin x + 2 sin 7x and
∂u
(x, 0) = 0 for all 0 ≤ x ≤ π.
∂t
2. Use separation of variables to find the solution to the heat equation
∂u
∂2u
=
2
∂x
∂t
over the range 0 < x < L, t > 0, with the perfectly-insulated boundary conditions
∂u
∂u
(0, t) =
(L, t) = 0,
∂x
∂x
and the initial condition u(x, 0) = f (x), 0 < x < L.
Find the particular solutions for the cases
1, 0 < x < L/2
(i) f (x) = x2 , (ii) f (x) =
0, L/2 < x < L.
(iii) f (x) =
1
cos
2
2πx
L
.
3. Find the solution of Laplace’s equation
∂2u ∂2u
+ 2 = 0,
∂x2
∂y
in the rectangle 0 ≤ x ≤ 2, 0 ≤ y ≤ 4, by the method of separation of variables, where:
(i) u
(ii) u
= 1 on the upper side and zero on the other three sides;
= sin(πx/2) on the upper side and zero on the other three sides.
4. It can be shown that the small vertical vibrations of a uniform beam are governed by
∂4u
∂2u
+ c2 4 = 0.
2
∂t
∂x
Seek a separable solution of the form u = X(x)T (t) and show that
X
T
=
=
A cos βx + B sin βx + C cosh βx + D sinh βx,
a cos cβ 2 t + b sin cβ 2 t,
where A, B, C, D, a, b, β are arbitrary constants. Hence find the solution subject to the boundary conditions
∂2u
∂2u
(0, t) =
(L, t) = 0,
u(0, t) = u(L, t) =
2
∂x
∂x2
and the initial conditions
∂u
u(x, 0) = f (x),
(x, 0) = 0, (0 < x < L).
∂t
Find the dependence on n of the frequency of the nth mode - how does this compare with that for the
wave equation in Q1?
A. G. Walton
M2AA2 Multivariable Calculus
2
Sheet 7 Answers
1. u = sin x cos
t + 2 sin 7x cos 7t.
RL
P∞
2. u = 12 a0 + n=1 an cos(nπx/L) exp(−n2 π 2 t/L2 ); an =(2/L) 0 f (x) cos(nπx/L) dx for n = 0, 1, 2, . . .
2 P∞ (−1)n
2
2 2
nπx
(i) u = L3 + 2L
exp − nLπ2 t ;
n=1 n2 cos
π
L
2 2 P∞
(ii) u = 12 + π2 n=1 sin(nπ/2)
cos nπx
exp − nLπ2 t ;
L
n
4π 2
t
.
(iii) u = 12 cos 2πx
exp
−
2
L
L
P∞
4
πx
3. (i) u = π m=1 sin((2m − 1) 2 ) sinh((2m − 1) πy
2 )/((2m − 1) sinh(2(2m − 1)π));
πy
(ii) u = sin πx
sinh
/
sinh(2π).
2
2
RL
P∞
4. u = n=1 Bn sin(nπx/L) cos(n2 π 2 ct/L2 ) with Bn = (2/L) 0 f (x) sin(nπx/L) dx for n = 1, 2, . . . .
The frequency ∝ n2 while in Q1 the frequency is proportional to n.
A. G. Walton
M2AA2 Multivariable Calculus
1
Problem Sheet 8
1. The function y(x, t) satisfies the one-dimensional wave equation
∂2y
∂2y
= c2 2 for 0 < x < ∞, t > 0
2
∂t
∂x
and is subject to
∂y
(0, t) = 0 for t ≥ 0,
∂x
∂y
y(x, 0) = 0,
(x, 0) = g(x) for 0 < x < ∞.
∂t
By taking a suitable Fourier transform show that the solution can be written in the form
Z ∞
2
gbc (ω)
cos(ωx) sin(ωct) dω.
y(x, t) =
πc 0
ω
2. Consider the solution of the heat equation
∂2u
∂u
=κ 2
∂t
∂x
over −∞ < x < ∞, t > 0 with the initial temperature distribution
u(x, 0) = e−|x| .
By taking a Fourier transform in x show that the solution is
Z
2 ∞ cos(ωx) −ω2 κt
u(x, t) =
e
dω.
π 0 1 + ω2
3. (i) Suppose we wish to find a bounded solution of the 2D Laplace’s equation
∂2φ ∂2φ
+ 2 =0
∂x2
∂y
over the half-space −∞ < x < ∞, y > 0, subject to the Dirichlet condition
φ(x, 0) = p(x).
Use the method of Fourier transforms to show that
Z
y ∞
p(ξ)
φ(x, y) =
dξ.
π −∞ (x − ξ)2 + y 2
(ii) Now consider the equivalent Neumann problem:
∂u
∂2u ∂2u
(x, 0) = q(x), u → 0 as x2 + y 2 → ∞.
+ 2 = 0,
2
∂x
∂y
∂y
Use the substitution v = ∂u/∂y to reduce this to a Dirichlet problem. Hence show that the solution is
Z ∞
1
ln((x − ξ)2 + y 2 ) q(ξ) dξ.
u(x, y) =
2π −∞
4. Let V be a bounded volume with surface S0 . The volume contains N holes, each with a surface
Si , i = 1, . . . , N. Suppose that
∇2 φ = f (r) in V
with Neumann boundary conditions
∂φ
= qi (r) on each Si (i = 0, . . . , N ).
∂n
Prove that the solution for φ is unique, up to an additive constant.
A. G. Walton
M2AA2 Multivariable Calculus
5. We wish to solve
∇2 φ = f (r)
in a bounded volume V subject to
∂φ
= q(r)
∂n
on the boundary S of V. By considering the Green’s function that satisfies
∇2 G = δ(r − r0 ) in V, ∂G/∂n = 0 on S,
show that the solution for φ can be written in the form
Z
Z
φ(r0 ) =
G(r; r0 )f (r) dV −
q(r)G(r; r0 ) dS + constant.
V
S
6. Verify that the solution for G which satisfies
∇2 G = δ(r − r0 ) for z > 0 with ∂G/∂z = 0 on z = 0
is given by
1
1
−
,
4π |r − r0 | 4π |r − r00 |
where r0 = (x0 , y0 , z0 ) and r00 = (x0 , y0 , −z0 ). Use this Green’s function to show that the solution of
G(r; r0 ) = −
∇2 φ = 0 for z > 0,
can be written as
φ(r0 ) =
If q has the specific form
Z
q(x, y) =
with q0 , R constants, show that
∂φ
= q(x, y) on z = 0,
∂z
q(x, y)G(r; r0 ) dx dy.
z=0
q0 , x2 + y 2 ≤ R2 ,
0, x2 + y 2 > R2 ,
φ(0, 0, z) = −q0 ((R2 + z 2 )1/2 − z).
7. Show that in two dimensions, a Green’s function that satisfies
∇2 G = δ(r)
is given by
G=
1
ln |r| .
2π
Hence deduce that the corresponding solution to
∇2 G = δ(r − r0 )
that holds in the upper half-plane y > 0, −∞ < x < ∞, and satisfies
G = 0 on y = 0,
is given by
1
1
ln |r − r0 | −
ln |r − r00 | ,
2π
2π
where r00 = (x0 , −y0 ) and r0 = (x0 , y0 ). Use this Green’s function to show that the solution of
G=
∂2φ ∂2φ
+ 2 = 0 for y > 0, φ(x, 0) = p(x)
∂x2
∂y
is given by
y
φ(x, y) =
π
Z
∞
−∞
p(ξ)
dξ,
(x − ξ)2 + y 2
a solution obtained by a different approach in Q3(i).
2