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Physics 211: Introduction to Mathematical Methods in Physics
4 credits
Catalog Course Description:
Series and complex numbers, vectors, matrix algebra, and fluid dynamics, fluid flow and divergence, circulation and the curl, applications of first-order differential equations to physics problems numerical integration, numerical methods and second-order differential equations (Euler method, Runge-Kutta method), oscillatory motion, resonance, forced oscillations.
Prerequisites:
PHY 114 and MTH 142
Rationale:
This course has two major objectives. They are:
- to provide sufficient in depth coverage of the mathematical tools used in junior- and senior-level physics courses so that students can concentrate on the physics content of the courses without requiring extensive mathematical review
- to develop problem solving skills by emphasizing manipulation of mathematical concepts through applications to a variety of concrete problems.
This course is specifically intended for sophomore students with one year of calculus (or freshman with AP calculus from high school) to develop, in a short time, a basic competence in each of the many areas of mathematics needed in junior to senior-graduate courses in physics and engineering. One of the major difficulties students have in the upper class physics and engineering courses has been their deficiencies in the mathematical skills necessary to develop problem solving techniques in the physical sciences. In the past, in order to cover the range of mathematics required, physics and engineering students had to either learn more mathematics than a mathematic major or learn a few areas thoroughly and others from brief sketches in the science courses. The proposed course by emphasizing development of skills in the area of mathematics required for the sciences will allow the student to obtain much more insight in the upper class science courses.
Syllabus:
| Week |
Topic |
| 1 |
Vectors: Definition, addition, scaling and properties under rotations. |
| 2 |
Scalar products and Vector products. Matrices and determinants and use in Vector algebra. |
| 3 |
Time-dependent vectors, vector functions, application to motion. |
| 4 |
Scalar and Vector Fields, applications to fluids and electricity. |
| 5 |
Vector Differential Calculus, the gradient and its interpretation. |
| 6-7 |
Vector integral calculus, volume and surface integrals, the curl and divergence theorems. |
| 8 |
Application of Vector analysis to physical fields by utilizing the divergence theorem, Stokes Law, Gauss Law, for problems with electric and magnetic fields and fluid mechanics. |
| 9 |
Complex numbers, complex power series, application of complex numbers. |
| 10-11 |
Partial differentiation. Multivariable power series, chain rule, application to minimum and maximum problems, Lagrange multipliers. |
| 12-13 |
Fourier series. Periodic functions, Fourier co-efficient, application to sound and periodic phenomena. Fourier transforms and the definition and application of the Dirac delta function. |
| 13-14 |
Ordinary differential equations. First and second order equations. Application to physics problems such as mechanics and forced oscillations. Application of numerical integration techniques such as Euler method and Runge-Kutta method. |
Suggested Texts:
- Vectors in Physics and Engineering, A.V. Durrant, Chapman & Hall, 1996
- Mathematical Methods in the Physical Sciences, 2nd Ed., M. Boas, John Wiley and Sons
Additional reference: Introduction to Mathematical Methods in Physics, Glenn Fetcher, Wm. C Brown, 1994
Student Assessment:
Grade will be based on:
- Homework problem solutions 10%
- Short quizzes (weeks 3,6,9,12,14) 20%
- Mid-term (week 8) 30%
- Final exam. 40%
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