Roy K.K. Potential theory in applied geophysics
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1.6 Gauss’s Divergence Theorem 9
Fig. 1.10. Shows the divergence of a vector
Let us take the first integral on the right hand side. We can now integrate
the first integral with respect to x, i.e., along a strip of cross section dy dz
extending from P
1
to P
2
.(Fig. 1.10).
We thus obtain
v
∂A
x
∂x
dxdydz =
s
[A
x
(x
2
,y,z) −A
x
(x
1
,y,z)] dydz. (1.31)
Here (x
1
, y, z) and (x
2
, y, z) are r espectively the coordinates of P
1
and P
2
.At
P
1
,wehave
dy dz = −dS
x
,
and at P
2
dy dz = dS
x
. (1.32)
Because the direction of the surface vectors are in the opposite direction.
Therefore
v
∂A
x
∂x
dxdydz =
s
A
x
dS
x
. (1.33)
where the surface integral on the right hand side is evaluated on the whole
surface. This way we can get
v
∂A
y
∂y
dxdydz =
s
A
y
dS
y
and (1.34)
v
∂A
z
∂z
dxdydz =
s
A
z
dS
z
. (1.35)
10 1 Elements of Vector Analysis
If we add these three components, we get Gauss’s theorem
v
(∇.V) dv =
s
(A
x
dS
x
+A
y
dS
y
+A
z
dS
z
)=
s
A.dS. (1.36)
1.7 Line Integral
Let A be a vector field in a space and MN is a curve described in the sense
M to N. Let the continuous curve MN be subdivided into infinitesimal vector
elements (Fig. 1.11)
dl
1
, dl
2
, dl
3
−−−−−−−dl
n.
.
The sum of these scalar products, is
N
M
A
r
−→
dl
r
=
N
M
A.dl. (1.37)
This sum along the entire length of the curve is known as the l ine integral of
A along the curve M N. In terms of Cartesian coordinate system, we can write
N
M
A.dl =
N
M
(A
x
dx + A
y
dy + A
z
dz). (1.38)
LetAbethegradientofφ, a scalar function of po sition, then
A =
∇φ (1.39)
and
N
M
A.dl =
N
M
(∇φ)dl =
N
M
∂φ
∂x
dx +
∂φ
∂y
dy +
∂φ
∂z
dz
. (1.40)
Fig. 1.11. Shows the paths of movement for line integral of a vector
1.8 Curl of a Vector 11
Since
∂φ
∂x
dx +
∂φ
∂y
dy +
∂φ
∂z
dz = dφ, (1.41)
we get
N
M
A.dl =
N
M
dφ = φ
N
− φ
M
(1.42)
where the value of φ
M
and φ
N
are the values of φ at the points M and N.
Therefore, the line integral of the gradient of any scalar function of position
φ around a closed curve vanishes. Because if the curve is closed, the p oint M
and N are coincident and the line integral is equal to φ
M
−φ
N
and is equal to
zero. In other words
MON
A.dl +
NPM
A.dl =0.
Hence
MON
A.dl = −
NPM
A.dl. (1.43)
This concept of line integral with a vector function, which is a gradient of
another scalar function, is used later to define potential in a scalar potential
field.
1.8 Curl of a Vector
Curl or circulation of a vector operates on a vector and generates another
vector (Fig. 1.12). It is written as ∇×
A, i.e., it is a cross product of the
vector operator ∇ and a vector A. Curl of a vector can be expla i ned using
the concept of line integral. If A is a vector, the curl or rot of A (circulation
or rotation) is defined as the vector function of space obtained by taking the
vector product of the operator ∇ and
A and its direction is at right angles to
the original vec tor. It is written as ∇×
A.Sowecanwrite
curl
A =
∇×
A =
i
∂A
z
∂y
−
∂A
y
∂z
+
j
∂A
x
∂z
−
∂A
z
∂x
+
k
∂A
y
∂x
−
∂A
x
∂y
. (1.44)
It can be written in a matrix form as
∇×
A =
i
J
k
∂
∂x
∂
∂y
∂
∂z
A
x
A
y
A
z
. (1.45)
12 1 Elements of Vector Analysis
A
Curl A
Fig. 1.12. Shows the circulation of a vector
If A = ∇φ,then
∇×A=∇×(∇φ)=
i
∂
2
φ
∂y∂z
−
∂
2
φ
∂z∂y
+
j
∂
2
φ
∂z∂x
−
∂
2
φ
∂x∂z
+
k
∂
2
φ
∂x∂y
−
∂
2
φ
∂x∂y
=0. (1.46)
Curl of a vector is zero if that vector can be defined as a gradient of another
scalar function.
1.9 Line Integral in a Plane (Stoke’s Theorem)
To show the connection between line integral and curl of a vector, let us
compute the line integral of a vector field
A around an infinitesimal rectangle
of sides ∆x and ∆y lying in the xy plane as shown in the (Fig. 1.13).
Fig. 1.13. Shows the line integral in a plane to explain stokes theorem
1.9 Line Integral in a Plane (Stoke’s Theorem) 13
We can compute
A.dl around the rectangle writing down the contribu-
tions towards this integral from different sides as follows:
Along AB → A
x
∆x
Along BC →
A
y
+
∂A
y
∂x
∆x
∆y
Along CD →−
A
x
+
∂A
x
∂y
∆y
∆x (1.47)
Along DA →−A
y
∆y.
Here ∆x and ∆y are infinitesimally small. Adding the various contributions,
we obtain
ABCD
A.dl =
∂A
y
∂x
−
∂A
x
∂y
∆x∆y (1.48)
It can be written as
ABCD
A.dl =
∇×
A
z
ds
xy
(1.49)
where ∇×
A
z
is the z-component of the curl of
A and ds
xy
is the area of the
rectangle ABCD.
If we take a closed surface s in the xy plane (Fig. 1.14) and the space is
divided into several rectangular el ements of infinitesima lly small areas, the
sum of the line integrals of the various meshes is given by
Fig. 1.14. Shows that the vectors only on the boundary remains and the vectors
inside get cancelled
14 1 Elements of Vector Analysis
∞
r=1
r
A.dl =
∞
r=1
(∇×A)
z
ds
xy
. (1.50)
Contributions to the line integrals of the ad j oining meshes cancel each other.
Only the line integrals in the periphery remain.
Hence
∞
r=1
r
A.dl =
c
A.dl (1.51)
and
A.dl =
s
(∇×A)
z
ds
xy
. (1.52)
This rela t i on is valid for all the components. Therefore we can write
A.dl =
(∇×A)ds. (1.53)
This is Stoke’s theorem. It states that surface integral of a curl of a vector
is equal to the line integral of the vector itself. It is a mathematical tool to
convert surface integrals to line integrals and vice versa.
1.10 Successive Application of the Operator ∇
In vector analysis, for successive application of operator ∇, we can take the
vector ∇×
B for ∇×∇×
A.where
Bis∇×
A. If we expand this equation
in cartisian coordinate, we get
∇×
A =
i
j
k
∂
∂x
∂
∂y
∂
∂z
A
x
A
y
A
z
. (1.54)
Here A
x
,A
y
and A
z
are respectively the x, y and z components of A in a
cartisian coordinate. Equation (1.54) is
Curl
A =
i
∂A
z
∂y
−
∂A
y
∂z
+
j
∂A
x
∂z
−
∂A
z
∂x
+
k
∂A
y
∂x
−
∂A
x
∂y
(1.55)
and curl
Bis
∇×∇×
A =
i
j
k
∂
∂x
∂
∂y
∂
∂z
∂A
z
∂y
−
∂A
y
∂z
∂A
x
∂z
−
∂A
z
∂x
∂A
y
∂x
−
∂A
x
∂y
(1.56)
1.11 Important Relations in Vector Algebra 15
⇒
i
∂
∂y
∂A
y
∂x
−
∂A
x
∂y
−
∂
∂z
∂A
x
∂z
−
∂A
z
∂x
+
j
∂
∂z
∂A
z
∂y
−
∂A
y
∂z
−
∂
∂x
∂A
y
∂x
−
∂A
x
∂y
+
k
∂
∂x
∂A
x
∂z
−
∂A
z
∂x
−
∂
∂y
∂A
z
∂y
−
∂A
y
∂z
(1.57)
⇒
i
∂A
y
∂y∂x
−
∂
2
A
x
∂y
2
−
∂
2
A
x
∂z
2
+
∂
2
A
z
∂z∂x
+
j
∂
2
A
z
∂z∂y
−
∂
2
A
y
∂z
2
−
∂A
y
∂x
2
+
∂
2
A
x
∂x∂y
+
k
∂
2
A
y
∂y∂x
−
∂
2
A
z
∂x
2
−
∂
2
A
z
∂y
2
−
∂
2
A
y
∂y∂z
. (1.58)
The ith component can be written as
⇒
i
∂
∂x
∂A
x
∂x
+
∂A
y
∂y
+
∂A
z
∂z
−
∂
2
A
x
∂x
2
+
∂
2
A
x
∂y
2
+
∂
2
A
x
∂z
2
. (1.59)
Writing the
jthand
kthcomponent as in (1.59), we can write
∇×∇×
A = ∇
∇.
A
−∇
2
A
or curl curl
A = grad div
A −∇
2
A. (1.60)
Equation (1.60) is an important relation and is used quite often in electro-
magnetic theory.
1.11 Important Relations in Vector Algebra
Some important relations in vector algebra, needed in p otential theory, are
presented in this section. A couple of relations are derived in the text. The
other relations can be derived. They are
i)
a.
b ×c=
b.c ×a=c.a ×
b (1.61)
ii)
a ×
b ×c
=(a.c)
b −
a.
b
c (1.62)
iii)
a ×
b
.
c ×
d
= a.
b ×
c ×
d
(1.63)
= a.
b.
dc −
b.
c
d
=(a.c)
b.
d
−
a.
d
bc
16 1 Elements of Vector Analysis
iv)
a ×
b
×
c ×
d
=
a ×
b.
d
c −
a ×
b.c
d (1.64)
v)
∇(φ + ψ)=∇φ + ∇ψ (1.65)
vi)
∇(φψ)=φ∇ψ + ψ∇φ (1.66)
vii)
∇.
a+
b
=
∇.a+
∇.
b (1.67)
viii)
∇. (φa) = φ∇.a+
a.∇φ (1.68)
ix)
∇.
a ×
b
=
b.
∇.a −a.
∇×
b (1.69)
x)
∇×
a+
b
=
∇×a+
∇×
b (1.70)
xi)
∇×(φa) = φ
∇a+∇φ ×a (1.71)
xii)
∇
a.
b
=(a.∇)
b+
b.∇
a+
a ×
∇×
b
+
b × (∇×a)
(1.72)
xiii)
∇×
a ×
b
= a.∇.
b −
b∇.a+
b.∇
a − (a.∇)
b (1.73)
xiv)
∇×
∇×
A
= ∇∇.
A −∇
2
A (1.74)
xv)
∇×(∇φ) = 0 (1.75)
xvi)
∇.
∇×a=0. (1.76)
2
Introductory Remarks
In th is chapter the ba sic ideas of (a) fields and their classifications (ii) poten-
tials (iii) boundary value problems and boundary conditions (iv) dimensional-
ity of a geophysical problem (v) nature of a medium in earth science (isotropy
and anisotropy, homogeneity and inhomogeneity) (vi) Tensors (vi)Nature of
equations encountered in solving geophysical problems (vii) Areas of geo-
physics controlled by potential theory are introduced.
2.1 Field of Force
At any point in a medium, an unit mass or an unit charge or an unit magnetic
po l e experiences a certain force. This force will be a force of attraction in
the case of a gravitational field. It will be a force of attraction or repulsion
when two unit charges or two magnetic poles of opposite or same polarity
are brought close to each other. These forces are fields of forces (Figs. 2.1,
2.2, 2.3).
A body at a point external to a single bod y or a gro u p of bodies will
experience force(s) of attraction in a gravitational field. These forces will be
exerted by a b ody or a group of bodies on a mass placed at a point. Every
mass in the space is associated with a gravitational force of attraction. This
force has b oth a direction and a magnitude. For gravitational field, the force
of attraction will b e between two masses along the line joining the bodies
(Fig. 2.1). For electrostatic, magnetostatic and direct current flow fields, the
direction of a field will be tangential to any point of observation. These forces
are the fields of forces. These fields are either global fields or are man made
artificial local fields. Thus we arrive at a conception of a field of force.
These fields are used to quantitatively estimate some physical properties
at every point in a medium. Important physical fields used by geophysicists
are (i)Gravitational field, (ii) Magnetostatic field, (iii) Electromagnetic field,
(iv) Direct current flow field, (v) Electrostatic fi eld, (vi) Heat flow field, (vii)
Fluid flow field, (viii) Earth’s natural electromagnetic field etc. Ea ch of these
18 2 Introductory Remarks
Fig. 2.1. Shows the gravitational attraction at a point P due to the masses m
1
,m
2
,
m
3
,andm
4
Fig. 2.2. Shows the magnetic field due to a bar magnet and the forces of attraction
and repulsion in the vicinity of tw o unlike and like magnetic poles
Fig. 2.3. Shows the electrostatic field due to a point charge