Exercises

10.11 EXERCISES
∇Φ
=
∇·a
=
∇×a
=
∇2 Φ
=
1 ∂Φ
1 ∂Φ
1 ∂Φ
ê1 +
ê2 +
ê3
h1 ∂u1
h2 ∂u2
h3 ∂u3
∂
1
∂
∂
(h2 h3 a1 ) +
(h3 h1 a2 ) +
(h1 h2 a3 )
h1 h2 h3 ∂u1
∂u2
∂u3
h1 ê1 h2 ê2 h3 ê3 1 ∂
∂
∂ h1 h2 h3 ∂u1
∂u2
∂u3 ha
ha
ha 1
h1 h2 h3
1 1
∂
∂u1
2 2
h2 h3 ∂Φ
h1 ∂u1
3 3
+
∂
∂u2
h3 h1 ∂Φ
h2 ∂u2
+
∂
∂u3
h1 h2 ∂Φ
h3 ∂u3
Table 10.4 Vector operators in orthogonal curvilinear coordinates u1 , u2 , u3 .
Φ is a scalar field and a is a vector field.
Letting Φ = a1 h1 in (10.60) and substituting into the above equation, we find
ê2 ∂
ê3 ∂
(a1 h1 ) −
(a1 h1 ).
h3 h1 ∂u3
h1 h2 ∂u2
The corresponding analysis of ∇ × (a2 ê2 ) produces terms in ê3 and ê1 , whilst that of
∇ × (a3 ê3 ) produces terms in ê1 and ê2 . When the three results are added together, the
coefficients multiplying ê1 , ê2 and ê3 are the same as those obtained by writing out (10.62)
explicitly, thus proving the stated result. ∇ × (a1 ê1 ) =
The general expressions for the vector operators in orthogonal curvilinear
coordinates are shown for reference in table 10.4. The explicit results for cylindrical
and spherical polar coordinates, given in tables 10.2 and 10.3 respectively, are
obtained by substituting the appropriate set of scale factors in each case.
A discussion of the expressions for vector operators in tensor form, which
are valid even for non-orthogonal curvilinear coordinate systems, is given in
chapter 26.
10.11 Exercises
10.1
10.2
Evaluate the integral
a(ḃ · a + b · ȧ) + ȧ(b · a) − 2(ȧ · a)b − ḃ|a|2 dt
in which ȧ, ḃ are the derivatives of a, b with respect to t.
At time t = 0, the vectors E and B are given by E = E0 and B = B0 , where the
unit vectors, E0 and B0 are fixed and orthogonal. The equations of motion are
dE
= E0 + B × E0 ,
dt
dB
= B0 + E × B0 .
dt
Find E and B at a general time t, showing that after a long time the directions
of E and B have almost interchanged.
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VECTOR CALCULUS
10.3
The general equation of motion of a (non-relativistic) particle of mass m and
charge q when it is placed in a region where there is a magnetic field B and an
electric field E is
mr̈ = q(E + ṙ × B);
here r is the position of the particle at time t and ṙ = dr/dt, etc. Write this as
three separate equations in terms of the Cartesian components of the vectors
involved.
For the simple case of crossed uniform fields E = Ei, B = Bj, in which the
particle starts from the origin at t = 0 with ṙ = v0 k, find the equations of motion
and show the following:
(a) if v0 = E/B then the particle continues its initial motion;
(b) if v0 = 0 then the particle follows the space curve given in terms of the
parameter ξ by
mE
mE
x = 2 (1 − cos ξ),
y = 0,
z = 2 (ξ − sin ξ).
B q
B q
Interpret this curve geometrically and relate ξ to t. Show that the total
distance travelled by the particle after time t is given by
Bqt 2E t dt .
sin
B 0
2m 10.4
10.5
Use vector methods to find the maximum angle to the horizontal at which a stone
may be thrown so as to ensure that it is always moving away from the thrower.
If two systems of coordinates with a common origin O are rotating with respect
to each other, the measured accelerations differ in the two systems. Denoting
by r and r position vectors in frames OXY Z and OX Y Z , respectively, the
connection between the two is
r̈ = r̈ + ω̇ × r + 2ω × ṙ + ω × (ω × r),
where ω is the angular velocity vector of the rotation of OXY Z with respect to
OX Y Z (taken as fixed). The third term on the RHS is known as the Coriolis
acceleration, whilst the final term gives rise to a centrifugal force.
Consider the application of this result to the firing of a shell of mass m from
a stationary ship on the steadily rotating earth, working to the first order in
ω (= 7.3 × 10−5 rad s−1 ). If the shell is fired with velocity v at time t = 0 and only
reaches a height that is small compared with the radius of the earth, show that
its acceleration, as recorded on the ship, is given approximately by
r̈ = g − 2ω × (v + gt),
where mg is the weight of the shell measured on the ship’s deck.
The shell is fired at another stationary ship (a distance s away) and v is such
that the shell would have hit its target had there been no Coriolis effect.
(a) Show that without the Coriolis effect the time of flight of the shell would
have been τ = −2g · v/g 2 .
(b) Show further that when the shell actually hits the sea it is off-target by
approximately
1
2τ
[(g × ω) · v](gτ + v) − (ω × v)τ2 − (ω × g)τ3 .
g2
3
(c) Estimate the order of magnitude ∆ of this miss for a shell for which the
initial speed v is 300 m s−1 , firing close to its maximum range (v makes an
angle of π/4 with the vertical) in a northerly direction, whilst the ship is
stationed at latitude 45◦ North.
370
10.11 EXERCISES
10.6
10.7
Prove that for a space curve r = r(s), where s is the arc length measured along
the curve from a fixed point, the triple scalar product
3
dr d2 r
dr
× 2 · 3
ds ds
ds
at any point on the curve has the value κ2 τ, where κ is the curvature and τ the
torsion at that point.
For the twisted space curve y 3 + 27axz − 81a2 y = 0, given parametrically by
x = au(3 − u2 ),
y = 3au2 ,
z = au(3 + u2 ),
show that the following hold:
√
(a) ds/du = 3 2a(1 + u2 ), where s is the distance along the curve measured from
the origin;
(b) the
√ length of the curve from the origin to the Cartesian point (2a, 3a, 4a) is
4 2a;
(c) the radius of curvature at the point with parameter u is 3a(1 + u2 )2 ;
(d) the torsion τ and curvature κ at a general point are equal;
(e) any of the Frenet–Serret formulae that you have not already used directly
are satisfied.
10.8
10.9
The shape of the curving slip road joining two motorways, that cross at right
angles and are at vertical heights z = 0 and z = h, can be approximated by the
space curve
√
√
zπ zπ 2h
2h
r=
i+
j + zk.
ln cos
ln sin
π
2h
π
2h
Show that the radius of curvature ρ of the slip road is (2h/π) cosec (zπ/h) at
height z and that the torsion τ = −1/ρ. To shorten the algebra, set z = 2hθ/π
and use θ as the parameter.
In a magnetic field, field lines are curves to which the magnetic induction B is
everywhere tangential. By evaluating dB/ds, where s is the distance measured
along a field line, prove that the radius of curvature at any point on a line is
given by
ρ=
10.10
B3
.
|B × (B · ∇)B|
Find the areas of the given surfaces using parametric coordinates.
(a) Using the parameterisation x = u cos φ, y = u sin φ, z = u cot Ω, find the
sloping surface area of a right circular cone of semi-angle Ω whose base has
radius a. Verify that it is equal to 12 ×perimeter of the base ×slope height.
(b) Using the same parameterization as in (a) for x and y, and an appropriate
choice for z, find the surface area between the planes z = 0 and z = Z of
the paraboloid of revolution z = α(x2 + y 2 ).
10.11
Parameterising the hyperboloid
y2
z2
x2
+ 2 − 2 =1
2
a
b
c
by x = a cos θ sec φ, y = b sin θ sec φ, z = c tan φ, show that an area element on
its surface is
1/2
dS = sec2 φ c2 sec2 φ b2 cos2 θ + a2 sin2 θ + a2 b2 tan2 φ
dθ dφ.
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VECTOR CALCULUS
Use this formula to show that the area of the curved surface x2 + y 2 − z 2 = a2
between the planes z = 0 and z = 2a is
√
1
πa2 6 + √ sinh−1 2 2 .
2
10.12
For the function
z(x, y) = (x2 − y 2 )e−x
10.13
2 −y 2
,
find the location(s) at which the steepest gradient occurs. What are the magnitude
and direction of that gradient? The algebra involved is easier if plane polar
coordinates are used.
Verify by direct calculation that
∇ · (a × b) = b · (∇ × a) − a · (∇ × b).
10.14
In the following exercises, a, b and c are vector fields.
(a) Simplify
∇ × a(∇ · a) + a × [∇ × (∇ × a)] + a × ∇2 a.
(b) By explicitly writing out the terms in Cartesian coordinates, prove that
[c · (b · ∇) − b · (c · ∇)] a = (∇ × a) · (b × c).
(c) Prove that a × (∇ × a) = ∇( 21 a2 ) − (a · ∇)a.
10.15
Evaluate the Laplacian of the function
ψ(x, y, z) =
10.16
10.17
x2
zx2
+ y2 + z2
(a) directly in Cartesian coordinates, and (b) after changing to a spherical polar
coordinate system. Verify that, as they must, the two methods give the same
result.
Verify that (10.42) is valid for each component separately when a is the Cartesian
vector x2 y i + xyz j + z 2 y k, by showing that each side of the equation is equal to
z i + (2x + 2z) j + x k.
The (Maxwell) relationship between a time-independent magnetic field B and the
current density J (measured in SI units in A m−2 ) producing it,
∇ × B = µ0 J,
can be applied to a long cylinder of conducting ionised gas which, in cylindrical
polar coordinates, occupies the region ρ < a.
(a) Show that a uniform current density (0, C, 0) and a magnetic field (0, 0, B),
with B constant (= B0 ) for ρ > a and B = B(ρ) for ρ < a, are consistent
with this equation. Given that B(0) = 0 and that B is continuous at ρ = a,
obtain expressions for C and B(ρ) in terms of B0 and a.
(b) The magnetic field can be expressed as B = ∇ × A, where A is known as
the vector potential. Show that a suitable A that has only one non-vanishing
component, Aφ (ρ), can be found, and obtain explicit expressions for Aφ (ρ)
for both ρ < a and ρ > a. Like B, the vector potential is continuous at ρ = a.
(c) The gas pressure p(ρ) satisfies the hydrostatic equation ∇p = J × B and
vanishes at the outer wall of the cylinder. Find a general expression for p.
10.18
Evaluate the Laplacian of a vector field using two different coordinate systems
as follows.
372
10.11 EXERCISES
(a) For cylindrical polar coordinates ρ, φ, z, evaluate the derivatives of the three
unit vectors with respect to each of the coordinates, showing that only ∂êρ /∂φ
and ∂êφ /∂φ are non-zero.
(i) Hence evaluate ∇2 a when a is the vector êρ , i.e. a vector of unit magnitude
everywhere directed radially outwards and expressed by aρ = 1, aφ =
az = 0.
(ii) Note that it is trivially obvious that ∇ × a = 0 and hence that equation
(10.41) requires that ∇(∇ · a) = ∇2 a.
(iii) Evaluate ∇(∇ · a) and show that the latter equation holds, but that
[∇(∇ · a)]ρ = ∇2 aρ .
(b) Rework the same problem in Cartesian coordinates (where, as it happens,
the algebra is more complicated).
10.19
Maxwell’s equations for electromagnetism in free space (i.e. in the absence of
charges, currents and dielectric or magnetic media) can be written
(i) ∇ · B = 0,
(ii) ∇ · E = 0,
∂B
1 ∂E
(iii) ∇ × E +
= 0, (iv) ∇ × B − 2
= 0.
∂t
c ∂t
A vector A is defined by B = ∇ × A, and a scalar φ by E = −∇φ − ∂A/∂t. Show
that if the condition
1 ∂φ
(v) ∇ · A + 2
=0
c ∂t
is imposed (this is known as choosing the Lorentz gauge), then A and φ satisfy
wave equations as follows:
1 ∂2 φ
= 0,
c2 ∂t2
2
1 ∂ A
(vii) ∇2 A − 2 2 = 0.
c ∂t
The reader is invited to proceed as follows.
(vi) ∇2 φ −
(a) Verify that the expressions for B and E in terms of A and φ are consistent
with (i) and (iii).
(b) Substitute for E in (ii) and use the derivative with respect to time of (v) to
eliminate A from the resulting expression. Hence obtain (vi).
(c) Substitute for B and E in (iv) in terms of A and φ. Then use the gradient of
(v) to simplify the resulting equation and so obtain (vii).
10.20
In a description of the flow of a very viscous fluid that uses spherical polar
coordinates with axial symmetry, the components of the velocity field u are given
in terms of the stream function ψ by
1 ∂ψ
−1 ∂ψ
,
uθ =
.
r2 sin θ ∂θ
r sin θ ∂r
Find an explicit expression for the differential operator E defined by
ur =
Eψ = −(r sin θ)(∇ × u)φ .
The stream function satisfies the equation of motion E 2 ψ = 0 and, for the flow of
a fluid past a sphere, takes the form ψ(r, θ) = f(r) sin2 θ. Show that f(r) satisfies
the (ordinary) differential equation
r4 f (4) − 4r2 f + 8rf − 8f = 0.
373
VECTOR CALCULUS
10.21
Paraboloidal coordinates u, v, φ are defined in terms of Cartesian coordinates by
x = uv cos φ,
y = uv sin φ,
z = 12 (u2 − v 2 ).
Identify the coordinate surfaces in the u, v, φ system. Verify that each coordinate
surface (u = constant, say) intersects every coordinate surface on which one of
the other two coordinates (v, say) is constant. Show further that the system of
coordinates is an orthogonal one and determine its scale factors. Prove that the
u-component of ∇ × a is given by
1
1 ∂av
∂aφ
aφ
−
+
.
2
2
1/2
(u + v )
v
∂v
uv ∂φ
10.22
Non-orthogonal curvilinear coordinates are difficult to work with and should be
avoided if at all possible, but the following example is provided to illustrate the
content of section 10.10.
In a new coordinate system for the region of space in which the Cartesian
coordinate z satisfies z ≥ 0, the position of a point r is given by (α1 , α2 , R), where
α1 and α2 are respectively the cosines of the angles made by r with the x- and ycoordinate axes of a Cartesian system and R = |r|. The ranges are −1 ≤ αi ≤ 1,
0 ≤ R < ∞.
(a) Express r in terms of α1 , α2 , R and the unit Cartesian vectors i, j, k.
(b) Obtain expressions for the vectors ei (= ∂r/∂α1 , . . . ) and hence show that the
scale factors hi are given by
h1 =
R(1 − α22 )1/2
,
(1 − α21 − α22 )1/2
h2 =
R(1 − α21 )1/2
,
(1 − α21 − α22 )1/2
h3 = 1.
(c) Verify formally that the system is not an orthogonal one.
(d) Show that the volume element of the coordinate system is
dV =
R 2 dα1 dα2 dR
,
(1 − α21 − α22 )1/2
and demonstrate that this is always less than or equal to the corresponding
expression for an orthogonal curvilinear system.
(e) Calculate the expression for (ds)2 for the system, and show that it differs
from that for the corresponding orthogonal system by
2α1 α2 R 2
dα1 dα2 .
1 − α21 − α22
10.23
Hyperbolic coordinates u, v, φ are defined in terms of Cartesian coordinates by
x = cosh u cos v cos φ,
y = cosh u cos v sin φ,
z = sinh u sin v.
Sketch the coordinate curves in the φ = 0 plane, showing that far from the origin
they become concentric circles and radial lines. In particular, identify the curves
u = 0, v = 0, v = π/2 and v = π. Calculate the tangent vectors at a general
point, show that they are mutually orthogonal and deduce that the appropriate
scale factors are
hu = hv = (cosh2 u − cos2 v)1/2 ,
10.24
hφ = cosh u cos v.
Find the most general function ψ(u) of u only that satisfies Laplace’s equation
∇2 ψ = 0.
In a Cartesian system, A and B are the points (0, 0, −1) and (0, 0, 1) respectively.
In a new coordinate system a general point P is given by (u1 , u2 , u3 ) with
u1 = 12 (r1 + r2 ), u2 = 12 (r1 − r2 ), u3 = φ; here r1 and r2 are the distances AP and
BP and φ is the angle between the plane ABP and y = 0.
374