2.035: SPECIAL TOPICS IN MATHEMATICS WITH APPLICATIONS

6 unit subject  -- Spring 2007

This year, the subject 2.035 will cover selected topics from Linear Algebra and the Calculus of Variations. It will be aimed mainly (but not exclusively) at students aiming to study mechanics (solid mechanics, fluid mechanics, energy methods etc.), and the course will introduce some of the mathematical tools used in these subjects. Applications will be related mostly (but not exclusively) to the microstructures of crystalline solids.

• Instructor:  Rohan Abeyaratne, 3-173, x3-2201, rohan@mit.edu
• Term:  Spring Term 2007
• Prerequisites:  Matrices and Multivariable Calculus
• Time:  Tuesdays and/or Thursdays 11:00-12:30 (6 unit subject; 14 classes)
• Place:   Room 1-134

Texts:

1. J.K. Knowles, Linear Vector Spaces and Cartesian Tensors, Oxford, 1998
2. Rohan Abeyaratne, Lecture Notes on The Mechanics of Elastic Solids: Volume 1: A Brief Review of Some Mathematical Preliminaries, free e-book, 2006. Download from http://web.mit.edu/abeyaratne/lecture_notes.html

Tentative Course Outline:

Linear Algebra: (8 classes)

• Linear vector spaces
• Euclidean vector spaces
• Linear transformations
• Cartesian tensors

Calculus of Variations: (6 classes)

• First variation
• Second variation.
• Variational principles in mechanics.
• Approximate solutions.

Grading:         Midterm = 40%, Final Exam = 60%

Further references: (On reserve at Barker Library)

1. I.M. Gelfand, Lectures on Linear Algebra, Dover, 1989.
2. P. R. Halmos, Finite Dimensional Vector Spaces, Van Nostrand-Reinhold, 1958.
3. I.M. Gelfand and S.V. Fomin, Calculus of Variations, Prentice Hall, 1963.
4. M. Giaquinta and S. Hilderbrandt, Calculus of Variations I, Springer, 1996.
5. J.L. Troutman, Variational Calculus with Elementary Convexity, Springer-Verlag, 1983.

Assigned Readings and Problem Sets:

Lecture 1: (February 8)

• Read pages 1-8 (Knowles)
• Key concepts: Vector Space, Linear Independence, Dimension of a Vector Space, Basis for Vector Space, Components of a Vector
• Work problems 1.1 to 1.11 (Knowles)

Lecture 2: (February 13)

• Read pages 9-17 (Knowles)
• Key concepts: Scalar Product, Length of Vector, Distance between Vectors, Angle between Vectors, Orthonormal basis
• Work problems 1.12 to 1.20 (Knowles)

Lecture 3: (February 22)

• Read pages 18-20, 23-26 (Knowles)
• Key concepts: Linear Transformations, Invariant Subspace, Eigenvalue problem
• Work problems 2.1, 2.3, 2.6, 2.17 except questions about singular/non-singular/inverse transforms (Knowles)

Lecture 4: (February 27)

• Read Chapter 2 (Knowles)
• Key concepts: Null Space, Singular/Non-Singular Linear Transformations, Inverse, Components of a Linear Transformation
• Work problems 2.1-2.5, 2.8, 2.9, 2.11, 2.15-2.17

Lecture 5: (March 6)

• Read pages 27-32, 42-46 (Knowles)
• Key concepts: Components of a Linear Transformation, Components in Different Bases, Scalar Invariants, Cartesian Tensors, Symmetric Tensors, Skew-Symmetric Tensors
• Work problems 3.1-3.12 (Knowles)

Lecture 6: (March 13)

• Read pages 42-52 (except Tensor Products), 56-57 (Knowles)
• Key concepts: Eigenvalues of a symmetric tensor, Principal basis, Positive-Definite Tensor, Orthogonal Tensor, Proper/Improper Orthogonal Tensor
• Work problems 3.13-3.18, 3.20, 3.24-3.26 (Knowles)

Lecture 7: (March 15)

• Read pages 44, 57-59 (Knowles), Chapters 2 and 3 (Abeyaratne)
• Key concepts: Tensor Product of 2 Vectors, Polar Decomposition of a Non-Singular Tensor
• Work problems 3.3-3.7, 3.11-3.16, 3.22, 3.23 (Knowles)

Midterm Exam: (April 3, and April 3-5)

• Part-1: In-Class. Tuesday April 3, 11:00 AM - 12:30 PM. You can use your own handwritten class notes only.
• Part-2: Take home. Due Thursday April 5 at 11 AM. You can use Knowles and your own handwritten class notes only.
• Old Exam: Mid-Term-Exam-Part-1 , Mid-Term-Exam-Part-2
Current Exam: 2007-Mid-Term-Exam-Part-1 , 2007-Mid-Term-Exam-Part-2
• Solutions: 2007-Mid-Term-Solution-Pt-1 , 2007-Mid-Term-Solution-Pt-2

Final Exam: (Distributed May 8, due May 15)

• The final exam will be entirely take home.
• The exam will be given out in class on May 8.
• Your solutions should be turned in no later than 11 AM on May 15.
• All problems will be on Calculus of Variations.
• There will be 6 problems of which you can work any 5. You should not devote more than 2 hrs on any one problem.
• The final exam will count for 50% of your grade.
• 2007 Final Exam: 2007-FinalExam
• 2007 Final Exam Solutions: 2007-FinalExamSolution