This course introduces a variety of mathematical techniques used in solving many problems in the field of physics. The course begins with a review of linear Euclidean vector spaces and generalizes this to tensor analysis and to the higher dimensions of Hilbert space. Linear algebra and its formalisms will be developed and applied to linear vector spaces and applied to simple problems of relativity. The physical intuition behind vector calculus theorems such as the Divergence Theorem and Helmholtz Theorem will be emphasized. The approach of linear algebra as applied to variables will then be generalized to analytical functions. The course will then introduce the concepts of basis eigenfunctions, eigenvalues, normalization and orthogonality in the context of infinite-dimensional Hilbert Space. The different approaches to solving different types of differential equations appearing in all branches of physics will be studied. Legendre polynomials and Hermite polynomials will be introduced as naturally arising in solving the hydrogen atom problem and quantum mechanics problems. The powerful theory of Green's function will be introduced and applied to problems in electromagnetism. Throughout the course, examples and applications that have connections to other branches of physics (e.g. classical mechanics, relativity, quantum mechanics, etc.) will be employed extensively to give the student the sense of universal utility of these mathematical methods. Prequisite: MAT-353.
