everything.study
2.035

Special Topics in Mathematics with Applications: Linear Algebra and the Calculus of Variations

Graduate · Historical course

Spring 2007

This course forms an introduction to a selection of mathematical topics that are not covered in traditional mechanical engineering curricula, such as differential geometry, integral geometry, discrete computational geometry, graph theory, optimization techniques, calculus of variations and linear algebra. The topics covered in any particular year depend on the interest of the students and instructor. Emphasis is on basic ideas and on applications in mechanical engineering. This year, the subject focuses on selected topics from linear algebra and the calculus of variations. It is aimed mainly (but not exclusively) at students aiming to study mechanics (solid mechanics, fluid mechanics, energy methods etc.), and the course introduces some of the mathematical tools used in these subjects. Applications are related primarily (but not exclusively) to the microstructures of crystalline solids.

Readings

  • Knowles, J. K. Linear Vector Spaces and Cartesian Tensors . New York, NY: Oxford University Press, 1998. ISBN: 9780195112542.
    Listed reading
  • Abeyaratne, Rohan. A Brief Review of Some Mathematical Preliminaries . Volume 1 in Lecture Notes on the Mechanics of Elastic Solids . [Free e-book.] ↗
    Listed reading
  • Gel’fand, I. M. Lectures on Linear Algebra . New York, NY: Dover, 1989. ISBN: 9780486660820.
    Listed reading
  • Halmos, P. R. Finite Dimensional Vector Spaces . Princeton, NJ: Van Nostrand-Reinhold, 1958. ISBN: 9780387900933.
    Listed reading
  • Gel’fand, I. M., and S. V. Fomin. Calculus of Variations . Englewood Cliffs, NJ: Prentice Hall, 1963.
    Listed reading
  • Giaquinta, M., and S. Hilderbrandt. Calculus of Variations I . New York, NY: Springer, 1996. ISBN: 9780387506258.
    Listed reading
  • Troutman, J. L. Variational Calculus with Elementary Convexity . New York, NY: Springer-Verlag, 1983. ISBN: 9780387907710.
    Listed reading
  • LEC #: 1 · Topics: Vector space, linear independence, dimension of a vector space, basis for vector space, components of a vector · READINGS: pp. 1-8 · PRACTICE PROBLEMS: Problems 1.1-1.11
    Reading assignment
  • LEC #: 2 · Topics: Scalar product, length of vector, distance between vectors, angle between vectors, orthonormal basis · READINGS: pp. 9-17 · PRACTICE PROBLEMS: Problems 1.12-1.20
    Reading assignment
  • LEC #: 3 · Topics: Linear transformations, invariant subspace, eigenvalue problem · READINGS: pp. 18-20 and 23-26 · PRACTICE PROBLEMS: Problems 2.1, 2.3, 2.6, and 2.17 (except questions about singular/non-singular/inverse transforms)
    Reading assignment
  • LEC #: 4 · Topics: Null space, singular/non-singular linear transformations, inverse, components of a linear transformation · READINGS: Chapter 2 · PRACTICE PROBLEMS: Problems 2.1-2.5, 2.8, 2.9, 2.11, and 2.15-2.17
    Reading assignment
  • LEC #: 5 · Topics: Components of a linear transformation, components in different bases, scalar invariants, cartesian tensors, symmetric tensors, skew-symmetric tensors · READINGS: pp. 27-32 and 42-46 · PRACTICE PROBLEMS: Problems 3.1-3.12
    Reading assignment
  • LEC #: 6 · Topics: Eigenvalues of a symmetric tensor, principal basis, positive-definite tensor, orthogonal tensor, proper/improper, orthogonal tensor · READINGS: pp. 42-52 (except tensor products) and 56-57 · PRACTICE PROBLEMS: Problems 3.13-3.18, 3.20, and 3.24-3.26
    Reading assignment
  • Also read chapters 2 and 3 in Abeyaratne, Rohan. Lecture Notes on the Mechanics of Elastic Solids .
    Listed reading
Sources