Page 20 - Physical Principles of Sedimentary Basin Analysis

2 Preliminaries

z
x
y
y
x
z
(a) (b)
Figure 1.1. (a) A right-handed coordinate system with the z-axis pointing upwards. (b) A right-
handed coordinate system with the z-axis pointing downwards.

is simply written as
x · y = x i y i (1.4)
when using Einstein’s summation convention. Here is another example:

3

σ ij n j = σ ij n j (1.5)
j=1
which shows the summation over a pair of equal indices. The summation convention is
often very useful, but it may lead to confusion. For instance, it implies that K ii = K 11 +
K 22 + K 33 , which is the sum over the diagonal elements. If we want K ii to denote one
(single) diagonal element we have to state that explicitly. One pair of equal indices may
be replaced by another pair of equal indices because there is a summation over them – for
example K ii = K jj . There is never summation over x, y and z when they are used as
indices. It is always possible to use these indices as numbers, where x = 1, y = 2 and
z = 3. We therefore have that
T
n = (n x , n y , n z ) T is the same as n = (n 1 , n 2 , n 3 ) . (1.6)
An important point is the notation for dimensionless quantities. When depth z is scaled with
a characteristic depth h it is written ˆz = z/h. A hat above a symbol denotes a dimensionless
x
quantity. For example, dimensionless spatial coordinates, time and temperature are ˆ, ˆy, ˆz,
ˆ t and T .
ˆ

1.2 Further reading

Riley et al. (1998) and Kreyszig (2006) are two comprehensive guides to mathematical
methods for physics and engineering.