Yang X.-S. Introductory Mathematics for Earth Scientists
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Introductory
Mathematics for
Earth Scientists
Xin-She Yang
DUNEDIN
DUNEDIN
D U N E D I N
Introductory Mathematics for Earth Scientists
Dr Xin-She Yang
Any quantitative work in earth sciences requires mathematical analysis to a
certain degree and many mathematical methods are essential to the modelling
and analysis of the geological, geophysical and environmental processes widely
studied in earth sciences. In this book Dr Yang provides an introduction to
the fundamental mathematics that all earth scientists need. The book is self-
contained and provides an essential toolkit of basic mathematics for earth
scientists assuming no more than a standard secondary school level as its
starting point.
The topics of earth sciences are vast and multidisciplinary, and consequently
the mathematical tools required by its students are diverse and complex. Thus
the author strikes a fine balance between coverage and detail. Topics have been
selected to provide a concise but comprehensive introductory coverage of all
the major and popular mathematical methods. The book offers a ‘theorem-
free’ approach with an emphasis on practicality. With dozens of step-by-step
worked examples, the book is especially suitable for non-mathematicians and
geoscientists.
Topics include: binomial theorem
s
index notations
s
polynomials
s
sequences and series
s
trigonometry
s
spherical trigonometry
s
complex
numbers
s
vectors and matrices
s
ordinary differential equations
s
partial
differential equations
s
Fourier transforms
s
numerical methods
s
geostatistics.
Introductory Mathematics for Earth Scientists introduces a wide range of
fundamental and yet widely-used mathematical methods. It is designed for use
by undergraduate students, though postgraduate students will also find it a
helpful reference and aide memoire.
Xin-She Yang received his DPhil in applied mathematics from the University of Oxford; he is
a senior research fellow at the University of Cambridge.
INTRODUCTORY MATHEMATICS FOR EARTH SCIENTISTS XIN-SHE YANG
ISBN 9781906716004
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6.6 Applications 91
This is equivalent to stating that
f(x
0
+ h) = f(x
0
) + hf
0
(x
0
) +
h
2
2
f
00
(x). (6.38)
If we again approximate f
00
(x) using higher-order derivatives, we can
obtain the Taylor se ries expansions
f(x
0
+ h) = f(x
0
) + f
0
(x
0
)h +
f
00
(x
0
)
2!
h
2
+
f
000
(x
0
)
3!
h
3
+ ... +
f
(n)
(x
0
)
n!
h
n
+ R
n+1
(h), (6.39)
where f
(n)
is the nth derivative of f(x) and R
n+1
(h) is the error esti-
mation for this se ries. In theory, we can use as many terms as possible,
but in pr actice, the series converges very quickly and only a few terms
are sufficient. It is straightforward to verify that the exponential series
for e
x
is identical to the results given earlier. Now let us look at other
examples.
Example 6.8: Let us expand f (x) = sin x abo u t x
0
= 0. We know
that
f
0
(x) = cos x, f
00
(x) = −sin x, f
000
(x) = −cos x, ...,
or f
0
(0) = 1, f
00
(0) = 0, f
000
(0) = −1, f
0000
(0) = 0, ..., which means
that
sin x = sin 0 + xf
0
(0) +
f
00
(0)
2!
x
2
+
f
000
(0)
3!
x
3
+ ... = x −
x
3
3!
+
x
5
5!
+ ...,
where the angle x is in radians.
For example, we know that sin 30
◦
= sin
π
6
= 1/2. We now use the
expansion to estimate it for x = π/3 = 0.523598
sin
π
6
≈
π
6
−
(π/6)
3
3!
+
(π/6)
5
5!
= 0.523599 − 0 .02392 + 0.0000328 ≈ 0.5000021326,
which is very close to the true value 1/2.
6.6 Applications
6.6.1 Free-Air Gravity
Let us use direct differentiation to derive the gradient of gravity for
free-air gravity correction.
92 Chapter 6. Differentiation
crust
subductio n
upper
mantle
lower
mantle
mid-ocean
ridge
Figure 6.9: Mantle convection and subduction zone.
We know the gravity g on the Earth’s surface is
g =
GM
E
r
2
, (6.40)
where M
E
is the mass of the Earth, and r is the radius of the Earth. G
is the universal g ravitational constant. The gra dient of gravity is the
first derivative with respect to r; we have
dg
dr
= GM
E
d
dr
(r
−2
) = GM
E
(−2)r
−2−1
= −
2GM
E
r
3
= −
2g
r
. (6.41)
Since g = 9.8 m/s
2
and r = 6370 km = 6.37 × 10
6
m, we have
dg
dr
= −
2 × 9.8
6.37 × 10
6
≈ −3.08 × 10
−6
m/s
2
m = −0.308 mGal/m,
where we have used geophysical units: 1 mGal = 10
−3
Gal = 10
−5
m/s
2
and 1 Gal = 1 cm/s
2
.
Therefore, if we go up to a height h, the free-air correc tion, or the
difference between the gravity at h and that on the ground, should be
∆g
h
= −0.308h mGal/m, (6.42)
which is well-known in gravity survey where the average gradient −0.3086
mGal/m is often used.
6.6.2 Mantle Convection
The main driving mechanism for plate tectonics is mantle convection,
arising from the temperature gradient above a certain critical level
under the right conditions. In such largescale convection, hot up-
welling fluid goes up at the mid-ocean ridge and cold downgoing current
subducts at the subduction zone (see Fig . 6.9).
6.6 Applications 93
For vigorous convection, convecting fluid is virtually adiabatic and
the entropy S of the system can be considered as constant. Ther-
modynamics analysis suggests that the adiabatic thermal gradient at
constant entropy is given by
dT
dp
=
T α
ρc
p
, (6.43)
where T is the average temperature, p is the pressure and α is the
thermal expansion coefficient. ρ and c
p
are the density and specific
heat of the convecting fluid, respectively. If we are concerned with
vertical gra dient only, as in the case of mantle convection, temperature
varies with depth z. Using the chain rule of differentiation, we have
dT
dp
=
∂T
∂z
∂z
∂p
=
∂T
∂z
/
∂p
∂z
. (6.44)
Assuming that the pressure is hydrostatic p = ρgz so that dp/dz = ρg
where g is the a c c e leration due to gravity, and using (6.43), we have
the adiabatic thermal gradient
dT
dz
=
dT
dp
dp
dz
=
T α
ρc
p
dp
dz
=
T αg
c
p
. (6.45)
Using typical values for upper mantle T = 1600 K, α = 3 × 10
−5
K
−1
,
c
p
= 1000 J K
−1
kg
−1
and g = 9.8 m/s
2
, we have
dT
dz
=
1600 × 3 × 10
−5
× 9.8
1000
≈ 0.47 × 10
−3
K/m = 0.47 K/km. (6.46)
This means that the temperature change in the 3000 km mantle is just
0.47 × 3000 ≈ 140 0 K.
Fluid dynamics has demonstrated that a horizontal fluid layer heated
from below remains quiescent until the thermal gradient exceeds a crit-
ical value, resulting in cellular conve c tion, called Rayleigh-B´enard con-
vection. For mantle convection, the dimensionless Rayleigh number Ra
is defined by
Ra =
ρgα∆T d
3
κµ
=
gα∆T d
3
κν
, (6.47)
where ν = µ/ρ is the kinematic viscosity and µ is the a bsolute or
dynamic viscosity. κ is the thermal diffusivity, ∆T is the temperature
difference across the convection layer with a thickness of d, a nd ρ is
the average density of the mantle. Convection occurs if Ra is greater
than a critical value Ra
∗
≈ 27 π
4
/4 ≈ 657.5 derived from instability
analysis, that is Ra > Ra
∗
. Viscosity in mantle varies greatly from
10
19
∼ 10
20
Pa s in the upper mantle to 1 0
20
∼ 10
23
Pa s in the lower
mantle. Using the typical values µ = 10
21
Pa s, ∆T = 1400 K, ρ = 3700
94 Chapter 6. Differentiation
kg/m
3
, κ = 10
−6
m
2
/s, d = 3000 km= 3 ×10
6
m, and the values for α
and g given earlier, we have
Ra =
3700 × 9.8 × 3 × 10
−5
× 1400 × (3 × 10
6
)
3
10
−6
× 10
21
≈ 4 × 10
7
. (6.48)
Such a hig h Rayleigh number implies that mantle convection is rigorous.
For Rayleigh-B´enard convection, the horizontal size of a convection cell
is similar to its vertical thickness. However, many studies o n mantle
convection suggest that the horizo ntal s ize is about 10, 00 0 km while
the vertical size varies from 670 km to 3000 km. This implies that the
viscosity in the lower mantle should be much higher than that in the
upper mantle. The detailed mechanism of mantle convection is still
poorly understood, and it is still an active area of current research.
Using a simple boundary layer theory with appropriate approxi-
mations, we k now tha t the convection flow is more rigorous with an
averaged convec tion velocity v inside a boundary layer with a thickness
of w. The total buoyancy force of the downgoing current is approxi-
mately
F
b
= ρgα(
∆T
2
)wA, (6.49)
where ∆T/2 is the average temperature difference, and A is the hori-
zontal area of the bounda ry layer. Since the shear stress o f the layer is
τ = µ(v/w) where v/w is the strain rate, the total viscous drag force
is
F
d
= τA =
µv
w
A. (6.50)
In the steady-state convection, the two forces should be balanced F
b
=
F
d
, and we have
ρgα(
∆T
2
)wA =
µv
w
A, (6.51)
or
v =
ρgα∆T w
2
2µ
. (6.52)
Detailed boundary-layer analysis (Turcotte and Schuber t, 1982; Richards
et al., 2000) suggests that w =
p
κd/v. We now have the velocity of
mantle convection
v =
s
ρgα∆T κd
2µ
. (6.53)
Using the typical values given ear lier, we have
v=
r
3700×9.8×3×10
−5
×1400 × 10
−6
× (3 × 1 0
6
)
2 × 10
21
≈1.5×1 0
−9
m/s,
which is about 5 cm/year since 1.5 × 10
−9
(m/s) ×36 5 × 24 × 3600
(s/year) ≈ 5 cm/year. This example demonstrates that the use of ap-
propriate values for various parameters is crucially important to obtain
a realistic estimate.
Chapter 7
Integration
Integration is another important topic in calculus. It can be considered
as the reverse of differentiation. Loosely speaking, differentiation and
integratio n are two sides of the same coin. Integrals are important in
finding areas, solv ing differe ntial equations, signal processing and many
branches of earth sciences such as geodynamics. Later in this chapter,
we will use integration to study unit hydrograph and derive the formula
for Bouguer gravity corrections.
7.1 Integration
Differentiation is used to find the gradient for a given function. Now a
natural question is how to find the original function fo r a given gradient.
This is the integration process, which can be considered as the reverse
of the differentiation process. Since we know that
d sin x
dx
= cos x, (7.1)
that is the gradient of sin x is cos x, we can easily say that the original
function is sin x if we know its gradient is cos x. We can write
Z
cos x dx = sin x + C, (7.2)
where C is the constant of integration. Here
R
dx is the standard
notation showing the integration is with respect to x, and we usually
call this the integral. The function cos x is called the integrand.
The integration constant comes from the fact that a family of curves
shifted by a constant will have the same gradient at their correspond-
ing points (see Fig. 6.2). This means that the integration can be de-
termined up to an arbitrary constant. For this reason, we call it the
indefinite integral.
95
96 Chapter 7. Integration
y
Area=
R
b
a
f(x)dx
x=a x=b
F
0
(x)= f(x)
+
−
+
Figure 7.1: Integration and geometrical interpretation.
Integration is more complicated than differentiation in general. Even
when we know the derivative of a function, we have to be careful. For
example, we know tha t (x
n+1
)
0
= (n + 1)x
n
or (
1
n+1
x
n+1
)
0
= x
n
for
any n integers, so we can write
Z
x
n
dx =
1
n + 1
x
n+1
+ C. (7.3)
However, there is a possible problem when n = −1 because 1/(n + 1)
will become 1/0. In fact, the above integra l is valid for any n except
n = −1. When n = −1, we have
Z
1
x
dx = ln x + C. (7.4)
If we know that the gradient of a function F (x) is f(x) or F
0
(x) =
f(x), it is pos sible and sometimes useful to express where the integra-
tion starts and ends, and we often write
Z
b
a
f(x)dx =
h
F (x)
i
b
a
= F (b) − F (a). (7.5)
Here a is called the lower limit of the integration, while b is the upper
limit of the integration. In this c ase, the constant of integration has
dropped out because the integral can be determined accurately. The
integral beco mes a definite integral and it corresponds to the area under
a curve f(x) from a to b ≥ a (see Fig. 7.1).
It is worth pointing out that the area in Fig. 7.1 may have different
signs depending on whether the curve f (x) is above or below the x-
axis. If f(x) > 0, then that section of area is positive, otherwise, it is
negative.
From (7.5), if we interchange the limits of the integral, we have
Z
a
b
f(x)dx=
h
F (x)
i
a
b
=F (a)−F (b)=−(F (b)−F (a))=−
Z
b
a
f(x)dx. (7.6)
7.1 Integration 97
y
+
−
−a
a
f(−x)=−f(x)
f(−x)=f(x)
−a a
Figure 7.2: Integrals of odd and even functions.
In addition, since differentiation is linear, that is [αf (x) + βg(x)]
0
=
αf
0
(x) + βg
0
(x) where α and β are two real constants, it is easy to
understand that integration is also a linear ope rator. We have
Z
b
a
[αf(x) + βg(x)]dx = α
Z
b
a
f(x)dx + β
Z
b
a
g(x)dx. (7.7)
Also if a ≤ c ≤ b, we have
Z
b
a
f(x)dx =
Z
c
a
f(x)dx +
Z
b
c
f(x)dx. (7.8)
This may beco me useful when dealing with certain integrals, especially
if we do not want the areas (positive and negative) to simply add to-
gether.
In the special case of an odd function f(−x) = −f(x), we have
Z
a
−a
f(x)dx =
Z
0
−a
f(x)dx +
Z
a
0
f(x)dx = −
Z
−a
0
f(x)dx +
Z
a
0
f(x)dx
=
Z
a
0
f(−x)dx +
Z
a
0
f(x)dx =
Z
a
0
[−f(x) + f (x)]dx = 0. (7.9)
The other way o f understanding this is that the a rea for x > 0 is
opposite in sign to the area for x < 0. So the total area or the integral
is zero.
Similarly, for an even function f(−x) = f(x), the area s for both
x > 0 and x < 0 are the same, so we have
Z
a
−a
f(x)dx = 2
Z
a
0
f(x)dx. (7.10)
Example 7.1: Let us try to compute the total area of the sha d e d
regions under the curve f(x) = x
2
− 5x + 4 shown in Fig. 7.3.
98 Chapter 7. Integration
1 2 3 4 5
2
4
−2
x
2
−5x+4
−
+
Fi
gure 7.3: Computing the total area of the shaded regions.
First, we can simply say that the area i s obviously the definite integral
A =
Z
4
0
f(x)dx =
Z
4
0
(x
2
− 5x + 4)dx.
From the basic linear properties of the integration, we have
A =
Z
4
0
x
2
dx − 5
Z
4
0
xdx + 4
Z
4
0
dx =
h
1
3
x
3
i
4
0
− 5
h
1
2
x
2
i
4
0
+ 4
h
x
i
4
0
=
1
3
× [4
3
− 0
3
] −
5
2
× [4
2
− 0
2
] + 4 × [4 − 0] =
64
3
− 40 + 16 = −
8
3
.
The total area is negative; this is not what we expected. The area s h ould
be positive, right? The geometrical meaning of the integral is the total
area, adding the positive areas (above the x-axis) and the negative (be low
the x-axis). However, what we wanted to calculate is si m p ly the total area
of the two shaded regions, not to include its signs. In this case, we have
to divide the integral into two intervals [0, 1] and [1, 4]. The area o f the
first region is
Z
1
0
f(x)dx =
1
3
× [1
3
− 0
3
] −
5
2
× [1
2
− 0
2
] + 4 × [1 − 0] =
11
6
.
Similarly, the area of the second region is
Z
4
1
f(x)dx =
1
3
× [4
3
− 1
3
] −
5
2
× [4
2
− 1
2
] + 4 × [4 − 1]
=
63
3
−
75
2
+ 12 = −
9
2
.
Here we only want the value of the area, so the area of the second region is
9/2. Therefore, the total area of the shaded regions is A =
11
6
+
9
2
= 6
1
3
.
7.1 Integration 99
The integrals of commonly used functions are listed in Table 7.1.
For many simple functions, we can easily write their integrals using
the table of differentiation. For more complicated functions, what ar e
the general techniques to obtain integrals? There are many techniques
such as integration by parts, method of substitution, co mplex integrals
and others. In the next section, we will briefly introduce integration
by parts.
Table 7.1: Integrals of common functions.
f(x)
R
f(x)dx
f(x)
R
f(x)dx
e
x
e
x
1
x
ln x
x
n
1
n+1
x
n+1
(n6=−1)
a
u
a
u
ln a
(a > 0)
ln x
x ln x − x log
a
x x log
a
x −
x
ln a
1
a
2
+x
2
1
a
tan
−1
x
a
1
a
2
−x
2
1
2a
ln
a+x
a−x
1
x
2
−a
2
1
2a
ln
x−a
x+a
1
√
a
2
−x
2
sin
−1
x
a
1
√
x
2
+a
2
sinh
−1
x
a
1
√
x
2
−a
2
cosh
−1
x
a
sin x
−cos x cos x sin x
sec
2
x
tan x csc
2
x −cot x
sinh x cosh x cosh x sinh x
tanh x
ln cosh x coth x ln sinh x
Example 7.2: Horton’s e q u a tion is an empirical relation ship conce rn-
ing the infiltration rates in groundwater flow. In the simplest case, the
infiltration rate f(t) at time t takes the form
f(t) = f
∗
+ (f
0
− f
∗
)e
−βt
,
where f
0
is the maximu m infiltration rate, and f
∗
is the infiltration rate
at the equilibrium after the soil has been saturated. Th e parameter β is
constant and τ = 1/β defines a characteristic timescale for infiltration.
In order to cal c u late the total volume of infiltration from time 0 to T ,
we take the integral
V =
Z
T
0
f(t)dt =
Z
T
0
[f
∗
+ (f
0
− f
∗
)e
−βt
]dt
= f
∗
T + (f
0
− f
∗
)
−1
β
e
−βt
T
0
= f
∗
T +
(f
0
− f
∗
)
β
(1 − e
−βT
).