Syllabus

This course is intended as both a refresher and a remedial course, giving an introduction to the following topics: first-order differential equations, second-order linear differential equations, linear algebra and systems of first-order linear equations, partial differential equations, Fourier series and boundary value problems. Example problems relevant to various aspects of chemistry will be used throughout the course.

Learning Outcomes

Upon successful completion of the course students will be able to:

1. Recognize the role of mathematics in various scientific fields.
2. Integrate knowledge from diverse fields such as calculus, linear algebra, differential equations, Fourier series to formulate and analyze models that arise in chemical reactions, in biology (population dynamics and predator-prey interactions) and mechanics and electricity in physics.
3. Apply methods from linear algebra to solve linear differential equations and systems of linear differential equations.
4. Apply a variety of different methods to solve special types of ordinary differential equations.
5. Apply Fourier series, and use the tools of Fourier analysis to solve partial differential equations (heat equation and wave equation).

Reading List

1. Arfken: Mathematical Methods for Physicists.
2. Boas: Mathematical Methods in the Physical Sciences.
3. Boyce and diPrima: Elementary differential equations and Boundary value problems, 7th edition.
4. Edwards and Penney: Elementary differential equations with Boundary value problems.
5. Mathews and Walker: Mathematical Methods of Physics.
6. Riley, Hobson and Bence: Mathematical Methods for Physics and Engineering Assignments.

Syllabus

This course is intended as both a refresher and a remedial course, giving an introduction to the following topics: first-order differential equations, second-order linear differential equations, linear algebra and systems of first-order linear equations, partial differential equations, Fourier series and boundary value problems. Example problems relevant to various aspects of chemistry will be used throughout the course.

1. Introduction to the complex numbers. Vector spaces, subspaces, linear combinations, span.
2. Matrices: operations, inverses. Gaussian elimination, rank of a matrix. Gauss-Seidel method for inverting matrices.
3. Linear independence, basis and dimension.
4. Solutions of systems of linear equations and the structure of the solution.
5. Determinants. Eigenvalues and eigenvectors, diagonalization of matrices. Jordan form.
6. Linear transformations: kernel and image, matrix of a transformation, changing bases.
7. Inner product spaces, orthogonality, Gram-Schmidt method. Hermitian and unitary matrices. Least squares solutions.

Learning Outcomes

Familiarity with matrix operations and techniques, solutions of linear systems of equations. Familiarity with basic concepts of linear algebra.

Reading List

1. Hoffman and Kunze, ”Linear Algebra”, Prentice-Hall 1971.
2. Lifschutz: Linear Algebra (Schaum series).
3. B. Noble and J.W. Daniel, Applied Linear Algebra, Prentice Hall, 1987.
4. G. Strang, Linear algebra and its applications, Brooks Cole, 2005.