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Lecture01

# Lecture01 - MATH 529 Mathematical Methods for Physical...

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MATH 529 – Mathematical Methods for Physical Sciences II Christoph Kirsch 0 Overview of the lecture This is the second half of a one year course on mathematical methods for physical sciences. The course web site is http://www.unc.edu/ ~ ckirsch/ MATH529 , which contains the course syllabus, course material such as these lecture notes, as well as the homework assignments. Please check this web site regularly for updates. The two topics covered in the lecture are partial differential equations complex analysis The course is based on the textbook Erwin Kreyszig: Advanced Engineering Mathematics. 9th Edi- tion; Wiley, 2006 We will basically go through chapters 12–18 of this textbook. It is presumed that students are familiar with multivariable calculus. Fourier series, integrals and transforms will be reviewed as needed. This lecture will connect naturally to the first half, MATH 528. Students who have not attended that lecture may notice a larger gap in the first few weeks. In addition to the weekly homework assignments, there will be two exams, a mid-term exam on March 3 (75 minutes, during regular class time), and the final exam on May 3, 8:00 AM (as set by the University Registrar). Classes of March 1 and April 26 will be review sessions in preparation of the respective exam. For the mid-term exam, you are allowed to use a summary of the lecture notes on three pages, and six pages for the final exam. Homework 1

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assignments and exams are graded and each counts for 1/3 of the total course grade. Please ask questions as they occur, and take advantage of the office hours, too! 0.1 Partial differential equations A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders. If the unknown function is of one variable, say u ( x ), we have an ordinary differential equation (ODE) of the form F ( x, u, u 0 , u 00 , . . . ) = 0 . (1) Solution techniques for these equations (series, Laplace transform, . . . ) have been discussed in MATH 528. For an unknown function of several variables, u ( x ), x = ( x 1 , . . . , x d ) R d , we have a partial differential equation (PDE) of the form F ( x , u, u, Hu , . . . ) = 0 , (2) (with a possibly different function F than in ( 1 ), of course) with the partial derivatives u = ∂u ∂x 1 . . . ∂u ∂x d , Hu = 2 u ∂x 2 1 . . . 2 u ∂x 1 ∂x d .
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