Last modified 9 May 2004 by jdavis@math.wisc.edu

Mathematical Methods in Geology, Spring 2004

Geology 376-002, meeting with Mathematics 491-001

We will be meeting Tuesday 9:15-10:30, Thursday 9:15-10:30, and Thursday 2:25-3:15. Most of the meetings will be lectures, but some will be discussions, labs, or group work sessions. All labs are in A320; all other meetings are in 212.

Junior Instructor: Joshua R. Davis

Senior Instructor: Basil Tikoff

Introduction

In a scientific paper, ideas are commonly explained in plain English and then rephrased in mathematical notation. Our comprehension is enhanced when we can understand the ideas in both languages. Unfortunately, not all readers read math well, and not all authors write math well.

In this course we will read math-intensive geological research papers, learning the math needed as we go along, filling in missing details and clarifying the presentation whenever possible. For each paper, I (Josh) will lecture on math for a day or two, and then a student will present the paper in detail, with the other students and Basil contributing geological insight. Students are asked to suggest papers that they find interesting and relevant!

By the end of the course, we will have reviewed the basics of a variety of mathematical concepts and techniques, including lots of calculus and linear algebra. Our review is necessarily cursory and ad hoc. But, since math is best learned by example, and our examples are pulled from geology, geology students should benefit. I also think that math students should benefit greatly, because even "pure" math is best learned with an eye on its historical development and scientific motivation.

Computer exercises will give us a chance to experiment with mathematical ideas. By the end of the course, students should be comfortable writing small programs to assist in scientific computations.

Expectations

Each student is expected to lead one class discussion. While preparing the paper, the student should meet with me at least once, so that we can discuss sticky math issues, correct errors in the paper, and agree on the important points. While talking about the paper, the presenter should use the blackboard to explain the math, and use the math to elucidate the geology. If there are not enough opportunities to speak on geology, then students should speak on math instead.

Beyond this, students are expected to read all of the papers and participate in class discussions. There will be occasional homework — either small computer programs or elementary exercises — so that students can practice basic concepts. There will be no exams.

Resources

The articles are listed in the syllabus. There is no textbook for the course. Those wanting mathematics references should pick up any textbook on calculus, linear algebra, or elementary differential equations. Relevant books, with which I'm familiar, include

Basil has mentioned the following, all of which are on reserve in the geology library. Also, over the years I have prepared the following densely packed, one-page summaries of elementary topics. They might be useful as a quick reminder of important points. Each is a PDF file of roughly 50 KB. But wait — there's more! Skylar pointed out this list of free math texts:

Syllabus

Calculus

  1. course introduction, limits and continuity, derivatives
  2. rules for derivatives, mean value theorem, L'Hopital's rule
  3. partial derivatives, antiderivatives, definite integrals
  4. fundamental theorem, improper integrals, Taylor series, Bessel functions, Homework 1: Calculus
  5. discussion (Steph): Ganguly and Tirone, Diffusion closure temperature and age of a mineral with arbitrary extent of diffusion: theoretical formulation and applications, Earth and Planetary Science Letters (1999) 131-140 (PDF)
  6. differentiability classes, analyticity, intuitive notion of vector space

Linear Algebra

  1. vectors, dot product, cross product, linear systems
  2. matrices, matrix arithmetic, inverses of 2x2 matrices
  3. Lab 1: Introduction to MATLAB
  4. inverses of 3x3 matrices, algebraic properties of matrices, linear systems via matrices, determinants, adjoints, inverses via adjoints
  5. discussion (Skylar): Ramsay and Graham, Strain variation in shear belts, Canadian Journal of Earth Sciences 7 (1970) 786-813
  6. discussion (continued), vector spaces, subspaces
  7. linear combinations, span, linear independence, basis, change of basis
  8. dimension, rank, linear transformations, change of basis, Homework 2: Vectors and Matrices
  9. Lab 1 (continued)
  10. matrix-transformation correspondence, kernel, range, eigenvalues and eigenvectors
  11. eigenstuff (continued), Jordan canonical form, diagonalization, transpose, symmetric matrices, trace and determinant as invariants

Ordinary Differential Equations

  1. hyperbolic functions, complex exponentiation, differential equations, diagonalizable first-order linear homogeneous ODE with constant coefficients
  2. discussion (Caroline): Ramberg, Particle paths, displacement and progressive strain applicable to rocks, Tectonophysics 28 (1975) 1-37
  3. group work: incremental strain limits to infinitesimal strain, separable ODE, first-order linear ODE
  4. group work: Homework 3: Linear ODE

Multivariable Calculus and Numerical Methods

  1. partial differentiation, gradient, directional derivatives
  2. vector fields, gradient, potential functions
  3. curl, divergence
  4. de Rham cohomology, Laplacian, heat equation, multiple integration
  5. group work: Homework 4: Vector Calculus
  6. Green's theorem, Stokes' theorem, divergence theorem, Newton's method
  7. approximating derivatives, Euler's method, Runge-Kutta method
  8. Lab 2: Numerical Methods
  9. discussion (Carrie): McDonald and Harbaugh, A modular three-dimensional finite-difference ground water flow model, in U.S. Geological Survey Techniques of Water-Resources Investigations, Book 6 (1988) Chapter A2

Fluid Dynamics

  1. fluid flow, streamlines, acceleration
  2. ideal fluids, Euler's equations
  3. viscous fluids, Navier-Stokes equations, simple shear flow example
  4. discussion (Wasinee): Middleton and Southard, Mechanics of sediment movement, second edition (1984) 303-326
  5. similarity solution of a diffusion equation arising from simple shear flow
  6. discussion (Preeya): Parker, Paola, and Leclair, Probabilistic Exner sediment continuity equation for mixtures with no active layer, Journal of Hydraulic Engineering 126 (2000) 818-826
  7. discussion (Eric): McKenzie, Finite deformation during fluid flow, Geophys. J. R. astr. Soc. 58 (1979) 689-715

Fourier Analysis and Thermodynamics

  1. trigonometric Fourier series, Fourier transform
  2. discussion (Emily): thermodynamics
  3. properties of Fourier transform, fundamental solution of heat equation
  4. discussion (Ben): Plummer and Phillips, A 2-D numerical model of snow/ice energy balance and ice flow for paleoclimatic interpretation of glacial geomorphic features, Quaternary Science Reviews 22 (2003) 1389-1406

Miscellany

  1. discussion (Drew): Dixon, An introduction to the global positioning system and some geological applications, Reviews of Geophysics 29, 2 (1991) 249-276, and Artru, Lognonne', Blanc, and Farges, Theory of acoustic coupling between solid Earth and its atmosphere, Proceedings OHP (2001)
  2. error functions, Hermite polynomials, Gram-Charlier series, Final Assessment

Other Ideas