Math Review for Geophysics

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Math Review for Geophysics

• By Jesse Lawrence
• Figures and outline from Stein and Wysession
  2003

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References

• The Fourier Transform and its Application, second
  edition, R.N. Bracewell, McGraw-Hill Book Co.,
  New York, 1978.
• An Introduction to Seismology, Earthquakes, and
  Earth Structure, S Stein and M. Wysession,
  Blackwell Publishing, Malden, 2003.

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Introduction

• Geodynamics is a math-heavy and computationally
  driven science. Without the proper math
  background it is impossible to make headway in
  geodynamics. Here, we will provide a cursory
  overview of the math that will be used in the
  remainder of this class.
• General Tools
• Complex Numbers
• Scalars and Vectors
• Spherical Coordinates
• Calculus
• Fourier Analysis

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Complex Numbers

• A complex number, z, has real and imaginary
  parts. Imaginary numbers are usually identified
  by an i, which is equal to ?-1. Real and
  Imaginary numbers can be visualized as existing
  in perpendicular (or orthogonal) planes.

z a ib rei?
z r(cos? i sin?)
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Complex Conjugate

• For every complex number, z, there is a complex
  conjugate, z. The complex conjugate is a useful
  tool in many calculations.

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Scalars Vectors

• Scalar a number describing a value that is
  independent of the coordinate system (e.g., Temp,
  Pressure, mass).
• Vector a series of scalars describing a value
  with a specific location in a coordinate system
  (e.g., GPS location).

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Coordinate Transformation

• The coordinates we use are defined to simplify
  calculations, but are otherwise arbitrary.

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Elementary Vector Operations

• Scalar, vector multiplication
• Sum of two vectors
• Unit vector (A vector with magnitude u1)

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Scalar Products
Also known as dot product or inner product
If a b
and
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Cross Product

• Also known as a vector product

Identities
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Rotation

• With a rotation vector, ?, at point r, we can
  find the linear velocity, v,

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Notation

• Index Notation
• Einstein summation convention

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Kronecker delta (?ij)

• Useful identity quantity
• So if i j,
• or if i ? j,

Note cos(90) 0 cos(0) 1
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Matrix Algebra

• Matrix Addition (MO NP)

• Matrix Multiplication (NO)

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Identity Matrix
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Transpose Matrix (AT)

• Swap the indices of the matrix (if C AT).
• Properties of Transpose Matrixes

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Determinant

• sgn is a function that determines the sign of an
  argument based on the series of values, returning
  -1, 0, or 1.
• for N 3
• So we can get

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Inverse Matrix (A-1)

• For a square matrix (N by N),
• The inverse matrix can be written in terms of a
  cofactor matrix, C, whose elements are,
• The inverse matrix can be written as,

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Matrix Inversion

• Using a linear system of

or
If A is square (MN)
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Rotation Matrix

• Recall that the dot product is a function of
  angle between two vectors.
• Transferring from one coordinate system to
  another, or simple rotation can be done through
  matrix algebra.
• To transfer,

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Example of Rotation

• 2D rotation about x3

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Fields

• Scalar Fields A series of scalar values within a
  field of locations (like temperature, density).
• Vector Fields A series of vectors within a field
  of location (like displacement, velocity,
  acceleration).

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Gradient

• The vector grad, ?, yields a first derivative of
  a field

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Directional Gradient

• The derivative of a field, ?, in a given
  direction, n, is given by,
• or,

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Divergence

• Derivatives of vector fields are simplified by
  the divergence

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Gauss Theorem

• The flow through a volume can be calculated from
  the flow through the surfaces of the volume

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Curl

• Curl describes the rotation or angular momentum
  of a vector field

Solar Wind
Earth
Magnetic Field
Permutation variable ?ijk 0 if ij, jk, or
ik. ?ijk 1 ijk are in order
(1,2,3),(3,1,2) ?ijk -1 ijk not in order
(3,2,1),(1,3,2)
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Stokes Theorem

• Analogous to Gauss Theorem for divergence,
  Stokes theorem relates the integral of curl of a
  vector on a surface, S, with some normal, n, to
  the integral of the vector field dotted with the
  tangent, t, for the line bounding the surface.

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Laplacian

• The divergence of the gradient of a field
• For a scalar field
• For a vector field

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Spherical Geometry

• For planetary calculations it is often useful to
  use spherical geometries

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Greater CircleDistance
Earthquake Location
Seismometer Location
The distance, ?, is given by the scalar product
of the two points. Or rearranged,
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Azimuth
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Vector Fields in Spherical Coordinates