3820 Mathematical Physics I focuses on applications of mathematical techniques to solve problems in physics. Vectors, vector calculus, matrices and tensors, coordinate systems and transformations, and summation notation are reviewed. Topics in complex numbers, functions and calculus are introduced, including branch cuts, differentiation, integration, Cauchy formula, series, residue theorem, and the gamma function. Other topics include differential equations using series solutions and separation of variables, and Fourier series of real and complex functions

PR: Mathematics 2260 (or 3260) and Mathematics 3202.

Philosophy is written in this grand book— I mean the universe— which stands continually open to our gaze, but it cannot be understood unless one first learns to comprehend the language in which it is written. It is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures, without which it is humanly impossible to understand a single word of it; without these, one is wandering about in a dark labyrinth (Galileo Galilei 1623).

Learning physics involves understanding basic concepts, such as those in first and second-year courses – classical mechanics and energy, electricity and magnetism, waves and optics, fluids and thermodynamics. Simple equations were introduced, like F=ma and KE=½mv2 that help us understand the concepts, and are used to solve problems that predict physical phenomenon. Knowledge of basic differentiation and integration techniques, linear algebra, vectors and differential equations are essential for solving many of the more interesting problems tackled in these courses. Third and fourth-year physics courses explore in more depth the concepts learned in elementary courses, and go beyond to introduce electromagnetic theory, quantum and statistical mechanics and more. Qualitative understanding is essential, but as Galileo advised us four hundred years ago, complete understanding requires the language of mathematics. PHYS 3820 provides a comprehensive foundation of the mathematics needed to understand and quantify the basic concepts in physics. Through extensive use of applications to physics problems, four of the major mathematical tools of the physicist are reviewed or introduced: Vector Calculus and Matrices, Complex Variables and Functions, Differential Equations, and Fourier Series and Transforms. The preferred textbook has recently been Mathematics for Physicists, S. M. Lea. (Thomson, 2004) where chapters 1-4 and 7 are mostly covered.