everything.study
3.016

Mathematics for Materials Scientists and Engineers

Undergraduate · Historical course

Fall 2005

Topics

  • Course Organization and Introduction to Mathematica®
  • Introduction to Mathematica®, Assignment and Evaluation, Rules and Replacement, Procedural and Functional Programming
  • Getting Started with Mathematica®
  • Mathematica® Graphics: Basic Plotting, Data, Two- and Three-dimensional Plotting, Graphics Primitives, Formatting
  • Mathematica®: Symbolic and Numeric Calculations, Linear Algebra, Roots of Equations
  • Mathematica®: Functional Programming, Packages, and File Input/Output
  • Symbolic Calculations and Plotting
  • Linear Algebra: Matrix Operations, Interpretations of Matrix Operations, Multiplication, Transposes, Index Notation
  • Linear Algebra: Solutions to Linear Systems of Equations, Determinants, Matrix Inverses, Linear Transformations and Vector Spaces
  • Solving Linear Systems of Equations
  • Complex Numbers: Complex Plane, Addition and Multiplication, Complex Conjugates, Polar Form of Complex Numbers, Powers and Roots, Exponentiation, Hyperbolic and Trigonometric Forms
  • Matrix Eigenvalues: Eigenvalue/Eigenvector Definitions, Invariants, Principal Directions and Values, Symmetric, Skew-symmetric, and Orthogonal Systems, Orthogonal Transformations
  • Hermitian Forms, Similar Matrices, Eigenvalue Basis, Diagonal Forms
  • File Input/Output, Plotting Data
  • Vector Calculus: Vector Algebra, Inner Products, Cross Products, Determinants as Triple Products, Derivatives of Vectors
  • Multi-variable Calculus: Curves and Arc Length, Differentials of Scalar Functions of Vector Arguments, Chain Rules for Several Variables, Change of Variable and Thermodynamic Notation, Gradients and Directional Derivatives
  • Statistics, Fitting Data, Error Analysis
  • Vector Differential Operations: Divergence and its Interpretation, Curl and its Interpretation
  • Path Integration: Integral over a Curve, Change of Variables, Multidimensional Integrals
  • Multidimensional Forms of the Fundamental Theorem of Calculus: Green’s Theorem in the Plane, Surface Representations and Integrals
  • Graphical Representations in Three and Higher Dimensions
  • Multi-variable Calculus: Triple Integrals and Divergence Theorem, Applications and Interpretation of the Divergence Theorem, Stokes’ Theorem.
  • Periodic Functions: Fourier Series, Interpretation of Fourier Coefficients, Convergence, Odd and Even Expansions
  • Fourier Theory: Complex Form of Fourier Series, Fourier Integrals, Fourier Cosine and Sine Transforms, The Fourier Transforms
  • Review of Mathematica® Functions and Graphics
  • Ordinary Differential Equations: Physical Interpretations, Geometrical Interpretations, Separable Equations
  • ODEs: Derivations for Simple Models, Exact Equations and Integrating Factors, The Bernoulli Equation
  • Higher Order Differential Equations: Homogeneous Second Order, Initial Value Problems, Second Order with Constant Coefficients, Solution Behavior
  • Differential Operators, Damped and Forced Harmonic Oscillators, Non-homogeneous Equations
  • Resonance Phenomena, Higher Order Equations, Beam Theory
  • Systems of Differential Equations, Linearization, Stable Points, Classification of Stable Points
  • Linear Differential Equations: Phase Plane Analysis and Visualization
  • Solutions to Ordinary Differential Equations
  • Solutions to Differential Equations: Legendre’s Equation, Orthogonality of Legendre Polynomials, Bessel’s Equation and Bessel Functions
  • Sturm-Louiville Problems: Eigenfunction, Orthogonal Functional Series, Eigenfunction Expansions

Readings

  • Kreyszig, Erwin. Advanced Engineering Mathematics . 8th ed. New York, NY: J.W. Wiley & Sons, 1999. ISBN: 9780471333753.
    Listed reading
  • Kreyszig, Erwin, and E. J. Norminton. Advanced Engineering Mathematics: Mathematica® Computer Manual . 8th ed. New York, NY: J. Wiley & Sons, 2001. ISBN: 9780471386698.
    Listed reading
  • SES #: L2 · TOPICS: Introduction to Mathematica®, Assignment and Evaluation, Rules and Replacement, Procedural and Functional Programming · READINGS: Lecture notes and Mathematica® notebook. ↗
    Reading assignment
  • SES #: Lab 1 · TOPICS: Getting Started with Mathematica® · READINGS: _Mathematica® Help Browser _Online Tutorial
    Reading assignment
  • SES #: L3 · TOPICS: Mathematica® Graphics: Basic Plotting, Data, Two- and Three-dimensional Plotting, Graphics Primitives, Formatting · READINGS: Lecture notes and Mathematica® notebook.
    Reading assignment
  • SES #: L4 · TOPICS: Mathematica®: Symbolic and Numeric Calculations, Linear Algebra, Roots of Equations · READINGS: Lecture notes and Mathematica® notebook.
    Reading assignment
  • SES #: L5 · TOPICS: Mathematica®: Functional Programming, Packages, and File Input/Output · READINGS: Lecture notes and Mathematica® notebook.
    Reading assignment
  • SES #: Lab 2 · TOPICS: Symbolic Calculations and Plotting · READINGS: _Mathematica® Help Browser _Kreyszig and Norminton: sections 1.4.2, 1.7.1. Functions: Integrate, Simplify, NIntegrate, Plot, Plot3D, ContourPlot.
    Reading assignment
  • SES #: L6 · TOPICS: Linear Algebra: Matrix Operations, Interpretations of Matrix Operations, Multiplication, Transposes, Index Notation · READINGS: Kreyszig. Sections 6.1, 6.2, 6.3, and 6.4.
    Reading assignment
  • SES #: L7 · TOPICS: Linear Algebra: Solutions to Linear Systems of Equations, Determinants, Matrix Inverses, Linear Transformations and Vector Spaces · READINGS: Kreyszig. Sections 6.5, 6.6, 6.7, and 6.8.
    Reading assignment
  • SES #: Lab 3 · TOPICS: Solving Linear Systems of Equations · READINGS: _Mathematica® Help Browser _Kreyszig and Norminton: section 1.8.3. Functions: Inverse, Transpose, Eigensystem, Matrix Multiplication.
    Reading assignment
  • SES #: L8 · TOPICS: Complex Numbers: Complex Plane, Addition and Multiplication, Complex Conjugates, Polar Form of Complex Numbers, Powers and Roots, Exponentiation, Hyperbolic and Trigonometric Forms · READINGS: Kreyszig. Sections 12.1, 12.2, 12.6, and 12.7.
    Reading assignment
  • SES #: L9 · TOPICS: Matrix Eigenvalues: Eigenvalue/Eigenvector Definitions, Invariants, Principal Directions and Values, Symmetric, Skew-symmetric, and Orthogonal Systems, Orthogonal Transformations · READINGS: Kreyszig. Sections 7.1, 7.2, and 7.3.
    Reading assignment
  • SES #: L10 · TOPICS: Hermitian Forms, Similar Matrices, Eigenvalue Basis, Diagonal Forms · READINGS: Kreyszig. Sections 7.4 and 7.5.
    Reading assignment
  • SES #: Lab 4 · TOPICS: File Input/Output, Plotting Data · READINGS: _Mathematica® Help Browser _Kreyszig and Norminton 2.12.7, 2.12.8. Functions: Dimensions, Append, AppendTo, Do, Mean, Standard Deviation, ListPlot, Table, Graphics ‘MultipleListPlot, Fit.
    Reading assignment
  • SES #: L11 · TOPICS: Vector Calculus: Vector Algebra, Inner Products, Cross Products, Determinants as Triple Products, Derivatives of Vectors · READINGS: Kreyszig. Sections 8.1, 8.2, 8.3, and 8.4.
    Reading assignment
  • SES #: L12 · TOPICS: Multi-variable Calculus: Curves and Arc Length, Differentials of Scalar Functions of Vector Arguments, Chain Rules for Several Variables, Change of Variable and Thermodynamic Notation, Gradients and Directional Derivatives · READINGS: Kreyszig. Sections 8.5, 8.8, and 8.9.
    Reading assignment
  • SES #: Lab 5 · TOPICS: Statistics, Fitting Data, Error Analysis · READINGS: _Mathematica® Help Browser _Kreyszig and Norminton: 3.8.2. Functions: Fit, FindFit; Package: Statistics ‘NonlinearFit.
    Reading assignment
  • SES #: L13 · TOPICS: Vector Differential Operations: Divergence and its Interpretation, Curl and its Interpretation · READINGS: Kreyszig. Sections 8.10 and 8.11.
    Reading assignment
  • SES #: L14 · TOPICS: Path Integration: Integral over a Curve, Change of Variables, Multidimensional Integrals · READINGS: Kreyszig. Sections 9.1, 9.2, and 9.3.
    Reading assignment
  • SES #: L15 · TOPICS: Multidimensional Forms of the Fundamental Theorem of Calculus: Green’s Theorem in the Plane, Surface Representations and Integrals · READINGS: Kreyszig. Sections 9.4, 9.5, 9.6, and 9.7.
    Reading assignment
  • SES #: Lab 6 · TOPICS: Graphical Representations in Three and Higher Dimensions · READINGS: _Mathematica® Help Browser _Kreyszig and Norminton: 1.9.1-1.9.7 and 1.9.9-1.9.11.
    Reading assignment
  • SES #: L16 · TOPICS: Multi-variable Calculus: Triple Integrals and Divergence Theorem, Applications and Interpretation of the Divergence Theorem, Stokes’ Theorem. · READINGS: Kreyszig. Sections 9.8 and 9.9.
    Reading assignment
  • SES #: L17 · TOPICS: Periodic Functions: Fourier Series, Interpretation of Fourier Coefficients, Convergence, Odd and Even Expansions · READINGS: Kreyszig. Sections 10.1, 10.2, 10.3, and 10.4.
    Reading assignment
  • SES #: L18 · TOPICS: Fourier Theory: Complex Form of Fourier Series, Fourier Integrals, Fourier Cosine and Sine Transforms, The Fourier Transforms · READINGS: Kreyszig. Sections 10.5, 10.8, 10.9, and 10.10.
    Reading assignment
  • Mathematica® Help Browser
    Listed reading
  • SES #: L19 · TOPICS: Ordinary Differential Equations: Physical Interpretations, Geometrical Interpretations, Separable Equations · READINGS: Kreyszig. Sections 1.1, 1.2, and 1.3.
    Reading assignment
  • SES #: L20 · TOPICS: ODEs: Derivations for Simple Models, Exact Equations and Integrating Factors, The Bernoulli Equation · READINGS: Kreyszig. Sections 1.4, 1.5, and 1.6.
    Reading assignment
  • SES #: L21 · TOPICS: Higher Order Differential Equations: Homogeneous Second Order, Initial Value Problems, Second Order with Constant Coefficients, Solution Behavior · READINGS: Kreyszig. Sections 2.1, 2.2, and 2.3.
    Reading assignment
  • SES #: L22 · TOPICS: Differential Operators, Damped and Forced Harmonic Oscillators, Non-homogeneous Equations · READINGS: Kreyszig. Sections 2.4, 2.5, and 2.8.
    Reading assignment
  • SES #: L23 · TOPICS: Resonance Phenomena, Higher Order Equations, Beam Theory · READINGS: Kreyszig. Sections 2.11 and 2.13 ( beam theory only ).
    Reading assignment
  • SES #: L24 · TOPICS: Systems of Differential Equations, Linearization, Stable Points, Classification of Stable Points · READINGS: Kreyszig. Sections 3.1 and 3.2
    Reading assignment
  • SES #: L25 · TOPICS: Linear Differential Equations: Phase Plane Analysis and Visualization · READINGS: Kreyszig. Sections 3.3 and 3.4.
    Reading assignment
  • SES #: L26 · TOPICS: Solutions to Differential Equations: Legendre’s Equation, Orthogonality of Legendre Polynomials, Bessel’s Equation and Bessel Functions · READINGS: Kreyszig. Sections 4.3, 4.5, and 4.6.
    Reading assignment
  • SES #: L27 · TOPICS: Sturm-Louiville Problems: Eigenfunction, Orthogonal Functional Series, Eigenfunction Expansions · READINGS: Kreyszig. Sections 4.7 and 4.8.
    Reading assignment
  • We will use a fairly general textbook on applied mathematics:
    Listed reading
  • Kreyszig, E. Advanced Engineering Mathematics . 8th ed. New York, NY: J.W. Wiley & Sons, 1999. ISBN: 9780471333753.
    Listed reading
  • We will also use this textbook’s Mathematica® supplement:
    Listed reading
Sources