gations numerically. One example is the change, both in exploration and in global seismology, from earth models whose
properties vary only with depth, to three-dimensional models
that are evaluated numerically.
The role of analytic solutions is also changing. In addition
to the traditional goal of providing exact solutions to simplified
problems, analytic solutions provide test cases for numerical
solutions of more complex problems. Analytic solutions can
also yield the insight needed to evaluate numerical results.
Along with the increase in the complexity of problems that
can be solved computationally comes an increase in the volume
of output. Fortunately, a parallel development has been the
increasing role of graphic output, often in color. The proverb
“A picture is worth a thousand words” may be unduly conservative in this context. A thousand words on a computer
might be 32,000 bits; graphic output often makes it possible to
visualize data with millions of bits.
Finally, software such as spreadsheets or programs with
sophisticated general mathematical capabilities often eliminates
the need to write programs for a specific application. In this
book, we do not assume that such software will be used for the
problems, although many could be done this way. We think
that programming without using such software gives a deeper
understanding of the underlying principles. Hence, in educational applications, we strongly favor programming, even if in
non-educational applications ease of use may favor sophisticated software.
Further reading
Many texts cover portions of the mathematical material summarized here.
Feynman (1982) discusses general issues of the relations between mathematics and science. Butkov (1968) and Menke and Abbott (1990) provide
introductions to many of these topics. Fung (1969), Hay (1953), Jeffreys
and Jeffreys (1950), and Marion (1970) treat vectors, vector transformations, and vector differential operators. Applied linear algebra texts such
as Franklin (1968) and Noble (1969) deal with the range of the subject
including numerical methods.
Articles by Hatton (1983a–d, 1984a,b, 1985) provide a broad and witty
introduction to computer science for geophysicists. Eckhouse and Morris
(1979) and Sloan (1980) cover topics in computer software, including the
representation of numbers and arithmetic operations. Kernighan and
Plauger (1976, 1978) discuss topics in programming style. Brooks (1975)
treats issues in the development and organization of computer software.
Numerical analysis texts like Froberg (1969) cover round-off and other
sources of error in numerical computations. Harkrider (1988) gives an entertaining anecdotal account of early (c.1960) computer usage in seismology.
The application of spherical geometry to the paths between an earthquake and a receiver, including the effects of the earth’s ellipticity are discussed by Ben-Menahem and Singh (1981) and Bullen and Bolt (1985).
The theory of the earth’s shape is treated by Cook (1973) and Jeffreys
(1976).
Further reading 473
Problems
1. Find the angle between the vectors (1, 4, 2) and (2, 3, 1).
2. Show, using index notation, that for the three-dimensional vectors
a, b, c:
(a) a × b is perpendicular to both a and b.
(b) | a × b | = | a || b | sin θ, where θ is the angle between the two
vectors.
(c) a · (b + c) = a · b + a · c.
(d) a × (b + c) = a × b + a × c.
(e) a · (b × c) = b · (c × a) = c · (a × b).
(f) a × (b × c) = b(a · c) − c(a · b).
3. Show that for arbitrary matrices A, B, and C:
(a) (AB)
T = B
T
A
T .
(b) (ABC)
T = C
T B
T
A
T .
4. Prove the following properties of determinants for the case of a
2 × 2 matrix:
(a) The determinant of a matrix equals the determinant of its
transpose.
(b) If two rows or columns of a matrix are interchanged, the determinant has the same absolute value, but its sign changes.
(c) If a multiple of one row (or column) of a matrix is added to
another row (or column), the determinant is unchanged.
(d) If two rows or columns of a matrix are the same, the
determinant is zero.
5. Express the determinant of a 3 × 3 matrix using the definition in
Eqn A.4.17.
6. Prove that if A has an inverse, the two solutions x and y satisfying
Ax = b and Ay = b are equal.
7. Find the inverse of the matrix
1 2
5 4
⎛
⎝
⎜
⎞
⎠
⎟
both by the cofactor method and by row operations. Check that
the solution is in fact the inverse.
8. Show that the inverse of a 2 × 2 matrix A is given by
A
A
a
a
a
a
− =
−
−
⎛
⎝
⎜
⎞
⎠
⎟
1
22
12
21
11
1
| |
.
9. Show that A, the transformation matrix for a rotation about the
ê 3 axis (Eqn A.5.9) satisfies A T A = I and is thus orthogonal.
10. Prove that the magnitude of a vector is preserved by an orthogonal
transformation.
11. Expand the determinant that give the eigenvalues of a 3 × 3 matrix
(Eqn A.5.19) and verify that the invariants (Eqn A.5.21) are the
coefficients of the characteristic polynomial.
12. Prove the following vector identities using index notation:
(a) For any vector field u(x), ∇ · (∇ × u) = 0.
(b) For any scalar function φ(x), ∇ × ∇φ = 0.
13. For the vector field u(x, y, z) = (3x 2 y 2 + z, 2x 3 y + 2y, x), find:
(a) ∇ · u.
(b) ∇ × u.
(c) ∇
2 u.
(d) A scalar field φ(x, y, z) such that u = ∇φ.
properties vary only with depth, to three-dimensional models
that are evaluated numerically.
The role of analytic solutions is also changing. In addition
to the traditional goal of providing exact solutions to simplified
problems, analytic solutions provide test cases for numerical
solutions of more complex problems. Analytic solutions can
also yield the insight needed to evaluate numerical results.
Along with the increase in the complexity of problems that
can be solved computationally comes an increase in the volume
of output. Fortunately, a parallel development has been the
increasing role of graphic output, often in color. The proverb
“A picture is worth a thousand words” may be unduly conservative in this context. A thousand words on a computer
might be 32,000 bits; graphic output often makes it possible to
visualize data with millions of bits.
Finally, software such as spreadsheets or programs with
sophisticated general mathematical capabilities often eliminates
the need to write programs for a specific application. In this
book, we do not assume that such software will be used for the
problems, although many could be done this way. We think
that programming without using such software gives a deeper
understanding of the underlying principles. Hence, in educational applications, we strongly favor programming, even if in
non-educational applications ease of use may favor sophisticated software.
Further reading
Many texts cover portions of the mathematical material summarized here.
Feynman (1982) discusses general issues of the relations between mathematics and science. Butkov (1968) and Menke and Abbott (1990) provide
introductions to many of these topics. Fung (1969), Hay (1953), Jeffreys
and Jeffreys (1950), and Marion (1970) treat vectors, vector transformations, and vector differential operators. Applied linear algebra texts such
as Franklin (1968) and Noble (1969) deal with the range of the subject
including numerical methods.
Articles by Hatton (1983a–d, 1984a,b, 1985) provide a broad and witty
introduction to computer science for geophysicists. Eckhouse and Morris
(1979) and Sloan (1980) cover topics in computer software, including the
representation of numbers and arithmetic operations. Kernighan and
Plauger (1976, 1978) discuss topics in programming style. Brooks (1975)
treats issues in the development and organization of computer software.
Numerical analysis texts like Froberg (1969) cover round-off and other
sources of error in numerical computations. Harkrider (1988) gives an entertaining anecdotal account of early (c.1960) computer usage in seismology.
The application of spherical geometry to the paths between an earthquake and a receiver, including the effects of the earth’s ellipticity are discussed by Ben-Menahem and Singh (1981) and Bullen and Bolt (1985).
The theory of the earth’s shape is treated by Cook (1973) and Jeffreys
(1976).
Further reading 473
Problems
1. Find the angle between the vectors (1, 4, 2) and (2, 3, 1).
2. Show, using index notation, that for the three-dimensional vectors
a, b, c:
(a) a × b is perpendicular to both a and b.
(b) | a × b | = | a || b | sin θ, where θ is the angle between the two
vectors.
(c) a · (b + c) = a · b + a · c.
(d) a × (b + c) = a × b + a × c.
(e) a · (b × c) = b · (c × a) = c · (a × b).
(f) a × (b × c) = b(a · c) − c(a · b).
3. Show that for arbitrary matrices A, B, and C:
(a) (AB)
T = B
T
A
T .
(b) (ABC)
T = C
T B
T
A
T .
4. Prove the following properties of determinants for the case of a
2 × 2 matrix:
(a) The determinant of a matrix equals the determinant of its
transpose.
(b) If two rows or columns of a matrix are interchanged, the determinant has the same absolute value, but its sign changes.
(c) If a multiple of one row (or column) of a matrix is added to
another row (or column), the determinant is unchanged.
(d) If two rows or columns of a matrix are the same, the
determinant is zero.
5. Express the determinant of a 3 × 3 matrix using the definition in
Eqn A.4.17.
6. Prove that if A has an inverse, the two solutions x and y satisfying
Ax = b and Ay = b are equal.
7. Find the inverse of the matrix
1 2
5 4
⎛
⎝
⎜
⎞
⎠
⎟
both by the cofactor method and by row operations. Check that
the solution is in fact the inverse.
8. Show that the inverse of a 2 × 2 matrix A is given by
A
A
a
a
a
a
− =
−
−
⎛
⎝
⎜
⎞
⎠
⎟
1
22
12
21
11
1
| |
.
9. Show that A, the transformation matrix for a rotation about the
ê 3 axis (Eqn A.5.9) satisfies A T A = I and is thus orthogonal.
10. Prove that the magnitude of a vector is preserved by an orthogonal
transformation.
11. Expand the determinant that give the eigenvalues of a 3 × 3 matrix
(Eqn A.5.19) and verify that the invariants (Eqn A.5.21) are the
coefficients of the characteristic polynomial.
12. Prove the following vector identities using index notation:
(a) For any vector field u(x), ∇ · (∇ × u) = 0.
(b) For any scalar function φ(x), ∇ × ∇φ = 0.
13. For the vector field u(x, y, z) = (3x 2 y 2 + z, 2x 3 y + 2y, x), find:
(a) ∇ · u.
(b) ∇ × u.
(c) ∇
2 u.
(d) A scalar field φ(x, y, z) such that u = ∇φ.