Course Outline

MATHEMATICAL PHYSICS (Lecture) Methods of formulating and solving physical problems that involve the use of mathematical tools such as coordinate systems and transformation, Fourier series and orthogonal functions, complex variables, ordinary differential equations, matrices and partial differential equations. This course is intended for sophomores and juniors, before they take quantum mechanics and other advanced physics courses. Prerequisites: Physics 222 or 252, Math 265.

Objectives

This course is designed to cover those parts of Mathematics which form the tools of the modern scientists in physics and closely related disciplines. This course provides a necessary preparation for advanced undergraduate physics courses and for graduate studies. Specific topical knowledge objectives are provided below in the course content. The acquisition of basic computer skills for handling the mathematics of physics is an integral learning objective of this course. The assessment of students' performance in this course takes into account the taxonomy of the cognitive domain. Relative lengthy homework assignments moderately emulate graduate work and the process of research.

Course Content

1. Polynomials and the Fundamental Theorem of Algebra:
Linear Equations, Determinants and Matrix Operations; Special Matrices (Orthogonal, Unitary, Symmetric, Hermitian, and Applications); Set of Linear Equations, Row Reduction, Determinants, Cramer's Rule, Linear Combinations, General Theorem of Sets of Linear Equations.The Coverage of 1., Herein, Must Precede that of 8. Below.

2. Series: General, Geometric, Harmonic, Alternating, and Power Series; Absolute, Conditional, and Uniform Convergences and Applications; Taylor's theorem and power series expansion, the uniqueness theorem and applications of the binomial theorem and of the uniform convergence for power series expansions;applications to include: the summation of many series using the computer; assessment of the validity of approximations.The Coverage of 2., herein, to precede that of 9. &10.

3. Complex Analysis: Real and Imaginary parts of Complex Number, Complex Plane, Complex Algebra. Complex Series, Circle of Convergence, Euler's Formula, Power and Roots of Complex Numbers, Trigonometric Functions, Hyperbolic Functions, Logarithms Functions. Coverage to precede that of 4-6 below.

4. Differential Equations: Generalities (Linear & non-linear, homogeneous & inhomogeneous, and ordinary and partial differential equations of nth order). Separation of Variables for Ordinary Differential Equations -- Applications to Linear First Order Differential Equations and to Second Order Equations (including Newton's); Special Case of 2nd Order Ordinary Differential Equations with Constant Coefficients and Applications to Oscillators. Coverage to follow that of 3.

5. Orthogonal Functions and Polynomials: General Survey, Legendre and other polynomials. Survey of Orthogonal Functions.[PHYS 411 treats selected general orthogonal functions.]Introduction to expansions in terms of orthogonal functions.The Coverage of 5. to precede that of 6.

6. Fourier Series: Simple Harmonic Motion, Average Value of a Function, Fourier Coefficients, Complex Form of Fourier Series, Even and Odd Functions. Applications to E&M potential functions. Coverage to follow that of 5.

7. Vector Analysis: Vector Multiplication, Differentiation of Vectors, Vector Fields (Gradient, Divergence, and Curl), Line Integrals, Green's Theorem, Divergence Theorem, Stoke's Theorem, Applications to E&M theory. Generation of Poisson, Laplace, and Wave equations from Maxwell's equation. Integral forms of the Maxwell equations. Role of source terms.

8. Coordinate Transformations: Linear Transformations, Orthogonal Transformations, Curvilinear Coordinates, Scale Factors and Basis Vectors for Orthogonal Systems, General Curvilinear Coordinates, Vector Operators in Orthogonal Curvilinear Coordinates; Cartesian Tensors and General Coordinate Systems. Applications to include: Schrodinger's equation for hydrogen, Poisson, Laplace, and other key physics equations in spherical polar and cylindrical coordinate systems. The Coverage of 8., herein, to follow that of 1.

9. Series Solution of Ordinary Differential Equations: The Frobenius Method. Applications to include tabulation and plotting (using the computer lab) of series solutions of key differential equations of physics.The Coverage of 9. to follow that of 4 & 2.

10. Partial Differential Equations: Laplace, Diffusion, Wave, and Poisson Equations and Others. Applications of the Frobenius method following the separation of variables. Applications to E&M theory. Identifications of special equations leading to various orthogonal polynomials.The Coverage of 10. to follow that of 5. & 9. Above.

NOTE: This course is a critical one for students' success in advanced physics courses and for most graduate courses in physics, engineering, and related disciplines.

 
Course Schedule

The course calendar is as dictated in the above course content. Some sections of the content require more time than others. The entire content listed above, however, is thoroughly covered.

Course Requirements

Prerequisites: Phys 222 or 252, Math 265 or equivalent ones.

Textbook: Mathematical Methods in the Physical Sciences by Mary L. Boas, Latest Edition, Publisher: John Wiley & Sons

Hardware & Software: The Computer Laboratory is in Room 113.Consult the Laboratory Manager or the Instructor for an updated listing of available software products.

Performance: Please refer to Evaluation Procedure below.

Accountability: If a student is in doubt as to the University's class attendance policy, she/he should consult the Catalog of SUBR and the dean of her/his college. Unless a valid and written excuse is produced, a zero will result for each missed quiz or test. Similarly, a zero will result for any homework not turned in on time.

Evaluation Procedure/Grading

Marks (grades) from homework, quizzes, tests, mid-term and final examinations are the bases for performance assessment in this course. A test or quiz should be expected any day of class. One homework assignment should be expected per week. Some emphasis is placed on relatively long homework assignments, in preparation for graduate work and research and development activities. The final examination will count for at least 30% of the semester grade. The mid-term examination will account for at least 20%. The algorithm for earning letter grades for the course follows: A: 90-100, B: 80-89, C: 70-79, D: 58-69, F: below 58.

Tautologies

One cannot consistently and correctly apply that which one does not know. Hence, timely and thorough learning from day one is paramount.

To go from understanding to knowing requires organizing, committing to memory the essentials (definitions, principles, laws, theorems, and skills), applying, practicing, and reviewing over time. The one who did not learn the definition of a polynomial and the fundamental theorem of algebra (FTA) will not, in a problem solving situation, recognize a polynomial of degree n nor will she/he realize that it always has n roots (or zeros). There is little to understand here while there is much to be known. Please see the attachment on Studying Successfully.