Introduction and theoretical background - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 74

3 − 4σ

R

1
Q +

2 ξ 3 x 3
Q 3 +

4 (1 − σ)(1 − 2σ)

u 1 = +

Q + x 3 + ξ 3

( x 1 − ξ 1 ) 2 1

x 3 + ξ 3 ) 2

3 − 4σ

Q 3 −

6 ξ 3 x 3
Q 5 −

4 (1 − σ)(1 − 2σ)

+

R 3 +

,

Q ( Q +

− ξ 2 ) 1

+ ξ 3 ) 2

3 − 4σ

Q 3 −

6 ξ 3 x 3 4 (1 − σ)(1 − 2σ)

= ( x 1 − ξ 1 )( x 2

R 3 +

Q 5 −

u 2 ,

Q ( Q

+ x 3

− ξ 1 ) x 3 − ξ 3
R 3

(3 − 4σ)( x 3 − ξ 3 )

Q 3

6 ξ 3 x 3 ( x 3 + ξ 3 )
Q 5

u 3 = ( x 1 + −

4 (1 − σ)(1 − 2σ)

+ . (1.529)

Q ( Q + x 3 + ξ 3 )

The displacement fields for the two Mindlin problems were used by Press (1965)
as the basis of the first demonstration, using dislocation theory, of the very extens-
ive nature of earthquake displacement fields.

1.5 Linear algebraic systems
Linear algebraic systems of equations are perhaps the most common systems of
equations arising in the study of Earth's dynamics. They are also extensively des-
cribed in the literature of mathematics and computation (see, for example, Press
et al. (1992), Wilkinson (1965)). The intention here is to provide a basic descrip-
tion, rather than repeat the exhaustive treatises available elsewhere.
In general, we will write the linear algebraic system of equations as

A x = b , (1.530)

where A is the coe cient matrix, x is the unknown vector, and b is the constant

vector. In the most common case, A is taken to be an n n square matrix, so the

vectors x and b are n -length. In addition to solving for the unknown vector x ,
we may wish to calculate the inverse, A − 1 ,ofthecoe

×

cient matrix, as well as its

determinant, .

For simplicity and clarity in expounding methods of solution for the three fore-
going problems, we will consider a 3

| A |

× 3 system. We will append the constant

vector and the unit matrix to the coe cient matrix in (1.530) to give

⎝ ⎠

a 11 a 12 a 13 b 1 100
a 21 a 22 a 23 b 2 010
a 31 a 32 a 33 b 3 001

, (1.531)

called the augmented matrix .
We may regard this as a system of equations with four successive constant vec-
tors. The usual elimination procedure for solving the linear system (1.530) may be