Mathematics for Materials Science and Engineers
MIT 3.016

An MIT undergraduate course covering mathematical techniques necessary for understanding of materials science and engineering topics such as energetics, materials structure and symmetry, materials response to applied fields, mechanics and physics of solids and soft materials. The class uses examples from the materials science and engineering core courses to introduce mathematical concepts and materials-related problem solving skills. Topics include linear algebra and orthonormal basis, eigenvalues and eigenvectors, quadratic forms, tensor operations, symmetry operations, calculus of several variables, introduction to complex analysis, ordinary and partial differential equations, theory of distributions, and fourier analysis and transformations, calculus of variations.

Fall 2012

W. Craig Carter
Office Hours: Mondays 9AM and usually available after lecture (Lecture weeks only) in 13-4053
Teaching Assistant
Rachel Zucker rzucker(at)
Office Hours: Tuesdays 3:30-4:30 (Lecture weeks only) in 13-4101
Laboratory Teaching Assistant
Ester Lomeli

3.016 Calendar

3.016 Calendar HTML (detailed, reading, lab topics, lecture topics)
3.016 Calendar PDF (detailed, reading, lab topics, lecture topics)

Problem Sets and Solutions

Laboratory Assignments

Previous Versions of these notes (2011)

Previous Versions of these notes (2010)

Previous Versions of these notes (2009)

All of 2007 notes (23 MB) with handy linked index

Notebooks Created in Recitation or in Discussions

Extra Material, Appendices to the Lecture Notes

Suggested Usage Paradigms and Elements of Style for Mathematica Beginners

Useful Links

2012 Lecture Notes and Notebooks:
The first column contains pdf with hyperlinks and is best if you are going to read the lecture on a computer screen; the other pdfs are intended for those who print. The Notebook contains an (unevaluated) Mathematica (8.0) notebook that goes with the lecture. The final column is html of the evaluated notebook. Animations will not be visible in the html

To download the notebook, right-click will work on many browsers

2012 Lecture Notes
Lecture Notes Mathematica
Date PDF (for reading onscreen) PDF (one-side print format) PDF (two-sides) Notebook HTML (of evaluated notebook)
Course Syllabus, Calendar, Grading Policies, Information Common Beginner Mistakes in Mathematica
Sept. 5 Lecture 1 (screen) Lecture 1 (print onesided) _Lecture 1 (print twosided) Lecture 1 (notebook) Lecture 1 (html-notebook)
Introduction to Mathematica I Assignments, expressions, operations on expressions, calculus and plotting, lists, rules, and replacement
Sept. 7 Lecture 2 (screen)Lecture 2 (print onesided) Lecture 2 (print twosided) Lecture 2 (notebook) Lecture 2 (html-notebook)
Introduction to Mathematica II Creating functions, patterns, immediate and delayed assignment, simple programs, plotting data, local variables
Sept. 10 Lecture 3 (screen)Lecture 3 (print onesided) Lecture 3 (print twosided) Lecture 3 (notebook) Lecture 3 (html-notebook)
Introduction to Mathematica III Simplifying and picking apart expressions, more calculus, solving equations, numerical operations, file input/output, using packages
Sept. 12 Lecture 4 (screen) Lecture 4 (print onesided) Lecture 4 (print twosided) Lecture 4 (notebook) Lecture 4 (html-notebook)
Introduction to Mathematica IV Graphics, graphics primitives, adjusting the behavior of plots, animations, manipulations, worked examples: Wulff Construction and Common Tangent Construction.
Sept. 14 Lecture 5 (screen) Lecture 5 (print onesided) Lecture 5 (print twosided) Lecture 5 (notebook) Lecture 5 (html-notebook)
Linear Algebra: vectors, matrices, multiplication, inversion and relations to linear systems of equations Matrix operations, inverses, determinants, matrix displays, threadable functions, linear dependence, eliminating redundant variables, rank, and null-space.
Sept. 17 Lecture 6 (screen) Lecture 6 (print onesided) Lecture 6 (print twosided) Lecture 6 (notebook) Lecture 6 (html-notebook)
Linear Algebra: Uniqueness, Vector Spaces, Rank, Nullity, Geometry and Symmetry. Solving linear and matrix equations, matrix inversion, properties of determinants, numerical precision: underflow, symmetry operations on polyhedra and crystal lattices
Sept. 19 Lecture 7 (screen) Lecture 7 (print onesided) Lecture 7 (print twosided) Lecture 7 (notebook) Lecture 7 (html-notebook)
Complex numbers and Operations, Euler's Formula Complex numbers, Operations on Complex Expressions, Polar representations of Complex numbers, numerical operations on complex numbers, complex roots to polynomial equations, examples of a bifurcation.
Sept. 24 Lecture 8 (screen) Lecture 8 (print onesided) Lecture 8 (print twosided) Lecture 8 (notebook) Lecture 8 (html-notebook)
Eigenvalues and Eigenvectors of Matrices, Special Matrix Forms Calculating eigenbasis, orthogonalization, similarity transformations, rotation of basis for matrices.
Sept. 26 Lecture 9 (screen) Lecture 9 (print onesided) Lecture 9 (print twosided) Lecture 9 (notebook) Lecture 9 (html-notebook)
Real Eigenvalue System, Transformations to Eigenbasis, Stress and Strain Tensors, General transformations of Tensors Calculating principle stresses and coordinate systems, Deriving Mohr's circle geometric representation of two-dimensional stress, Example of Mouse-Interaction Graphics.
Sept. 28 Lecture 10 (screen) Lecture 10 (print onesided) Lecture 10 (print twosided) Lecture 10 (notebook) Lecture 10 (html-notebook)
Vector products and norms, Space curves and their tangent vectors, Review of multivariable calculus, time-dependent scalar and gradient vector fields. Vector algebra, Visualization of space curves and vector tangents, particular solution to the diffusion equation and its visualization.
Oct. 10> Lecture 11 (screen) Lecture 11 (print onesided) Lecture 11 (print twosided) Lecture 11 (notebook) Lecture 11 (html-notebook)
Multivariable Calculus, Arclength, Curves and Curvature, Thermodynamics, Total and Partial Derivatives, Taylor Expansions Embedding curves in surfaces, calculating and inverting arclength, partial and total derivatives, series expansions in several variables, visualization of approximations at points of a surface.
Oct. 12 Lecture 12 (screen) Lecture 12 (print onesided) Lecture 12 (print twosided) Lecture 12 (notebook) Lecture 12 (html-notebook)
Differential Operations on Vectors and Vector Fields, Interpretation of the Curl. Scalar potentials and gradient fields, the Laplacian operator and visualization, coordinate transformations, Div, Grad, Curl in other coordinate systems, Geodesy.
Oct. 15 Lecture 13 (screen) Lecture 13 (print onesided) Lecture 13 (print twosided) Lecture 13 (notebook) Lecture 13 (html-notebook)
Path integrals, Path-Dependence, Path-Indepence in Thermodynamics, Maxwell Relations, Multi-dimensional integration, Derivatives of Definite Integrals, Jacobians, Change-of-Variables in Thermodynamics, Electrostatic Potential above a charged surface-patch. Examples of path integrals of vanishing and non-vanishing curl vector functions, vector functions with vanishing curl on a surface; Integrals over variable domains, numerical integration of a triangular surface-patch potential; Progress indicators for time-consuming computations, animation of contour plots.
Oct. 17 Lecture 14 (screen) Lecture 14 (print onesided) Lecture 14 (print twosided) Lecture 14 (notebook) Lecture 14 (html-notebook)
Surface Integrals and Higher-Dimensional Generalizations of the Fundamental Theorem of Calculus; Green's theorem; Interpretation of Curl; Representation of Surfaces; Local Coordinate Systems at Surfaces; Tangent Planes, Surface Normals, Curvature. Green's Theorem and Numerical Efficiency; Visualization of Graphs, Parametric Surfaces, and Level Sets; Floating Pixels, Distorted Images; Integration of Anisotropic Surface Energy.
Oct. 19 Lecture 15 (screen) Lecture 15 (print onesided) Lecture 15 (print twosided) Lecture 15 (notebook) Lecture 15 (html-notebook)
Higher Dimensional Integral Theorems, Interpretation of Divergence, Stokes' Theorem, Gauss' Law, Ampere's Law, Maxwell's Equations Calculating the Hamaker Potential from a Finite Cylindrical Rod: Reducing an "impossible" three dimensional integral to a one-dimensional numerical integration.
Oct. 29 Lecture 16 (screen) Lecture 16 (print onesided) Lecture 16 (print twosided) Lecture 16 (notebook) Lecture 16 (html-notebook)
Periodic Functions and Representations by Fourier Series, Odd and Even functions, Functional Basis, Functional Orthogonality, Complex Form of the Fourier Series Sound, Noise, and Music; The Mod function, Orthogonality of the trignometric functions; Calculating Fourier Coefficients and Truncated Fourier Series; Gibbs Phenomenon.
Oct. 31 Lecture 17 (screen) Lecture 17 (print onesided) Lecture 17 (print twosided) Lecture 17 (notebook) Lecture 17 (html-notebook)
Fourier Transforms and its Interpretation, Extension of Fourier Series to Infinite Wavelength, Higher dimensional Fourier Transforms, Transforms of Delta-Functions, Convolution Theorem Functional Basis, Functional Orthogonality, Complex Form of the Fourier Series Simulated Diffraction and Selected Area Diffraction from Simulated Atomic Scatters, Polycrystals, and Lattices with Thermal Vibrations.
Nov. 2 Lecture 18 (screen) Lecture 18 (print onesided) Lecture 18 (print twosided) Lecture 18 (notebook) Lecture 18 (html-notebook)
Introduction to Ordinary Differential Equations, Forward Differencing Methods, Geometry of Solutions to ODEs, Separable Equations Simple iteration of initial values, iteration sequences and trajectories, explicit forward difference, implicit forward difference, numerical stability, visual interpretation of ODE behavior, DSolve
Nov. 5 Lecture 19 (screen) Lecture 19 (print onesided) Lecture 19 (print twosided) Lecture 19 (notebook) Lecture 19 (html-notebook)
Linear Homogeneous and Heterogeneous First-Order ODEs, Models to ODEs: Grain Growth, Integrating Factors and Exact Differential Forms, Integrating Factors and Thermodynamics, Transforming Variables in an ODE: The Bernoulli Equation. General Solutions to General Linear Homogeneous and Heterogeneous First-Order ODEs, Transformation of Variables in ODEs, Numerical Solutions to Nonlinear First-Order ODEs, Extracting Solutions for Visualization from Numerical Solutions to ODEs.
Nov. 7 Lecture 20 (screen) Lecture 20 (print onesided) Lecture 20 (print twosided) Lecture 20 (notebook) Lecture 20 (html-notebook)
Second and Higher Order ODEs, Second-Order Differencing methods, Superposition of Solutions, Solution Function Basis: Linear Combinations of Basis Solutions, Homogeneous Second-Order ODEs with constant coefficients. Characterization of Solution Behavior, Determination of Constants from Boundary Conditions, the Beam Equation, Explicit Second-Order Differencing and its Visualization, Deriving Solutions to 2nd Order ODEs with constant coefficients, Building Up Graphics to Characterize Solution Behavior in terms of Constant Coefficients, Solving for Boundary Conditions, General Solutions to the Beam Equation for Common Loading and Boundary Condtions, Visualization of Beam Deflections, a Gratiutous Example of an Interactive Beam Equation Solver.
Nov. 9 Lecture 21 (screen) Lecture 21 (print onesided) Lecture 21 (print twosided) Lecture 21 (notebook) Lecture 21 (html-notebook)
Lniear and Integral Differential Operators, Fourier Transform Solutions to the Damped Harmonic Oscillator, Functionals, Variation of Paramters, Calculus of Variations, Geodesics and Minimum-Time Paths, Variational Derivatives. Using fourier transforms, rules for distributable functionals, solving the damped harmonic oscillator, approximating and calculating and visualizing the geodesic, computing shortest-time paths from a velocity model, using variational techniques, solving Euler's equation.
Nov. 21 Lecture 22 (screen) Lecture 22 (print onesided) Lecture 22 (print twosided) Lecture 22 (notebook) Lecture 22 (html-notebook)
Damped forced harmonic oscillator, resonance phenomena. Analysis of damped forced harmonic oscillators
Nov. 26 Lecture 23 (screen) Lecture 23 (print onesided) Lecture 23 (print twosided) Lecture 23 (notebook) Lecture 23 (html-notebook)
Systems of differential equations, linearization, stable points, classification of stable points Saddles, circles, focii
Nov. 28 Lecture 24 (screen) Lecture 24 (print onesided) Lecture 24 (print twosided) Lecture 24 (notebook) Lecture 24 (html-notebook)
Linear differential equations: phase plane analysis and visualization Phase planes and evolution of systems of ODEs
Nov. 30 Lecture 25 (screen) Lecture 25 (print onesided) Lecture 25 (print twosided) Lecture 25 (notebook) Lecture 25 (html-notebook)
to be announced to be announced
Dec. 5 Lecture 26 (screen) Lecture 26 (print onesided) Lecture 26 (print twosided) Lecture 26 (notebook) Lecture 26 (html-notebook)
to be announced to be announced
Dec. 7 Lecture 26 (screen) Lecture 26 (print onesided) Lecture 26 (print twosided) Lecture 26 (notebook) Lecture 26 (html-notebook)

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The general public is free to use these notes for educational purposes. Complete sets can be made available, please contact please if you copy these notes and distribute them in any way, please attribute properly. If you use these notes, please attribute the author; I would very much appreciate your comments, both positive and negative, about these notes. ---WCC