Phys209 & Phys210

What are these courses?

For a formal answer to this question, see

and

More informally, Phys209 introduces the differential equations and expands on how one can approach (and, if possible, solve) these objects whereas Phys210 introduces the vector calculus and the complex analysis. We are interested in these objects because most of the known physics is expressed in terms of differential equations whose solutions are organized using the language of vectors and complex numbers.

Who can/should take these courses?

This is a must course for any undergraduate student in the department of Physics (see the curriculum) and the department of Physics Education (see the curriculum). Assuming that you have taken a sufficient set of prerequisite courses (see Phys209 and Phys210), you should take these courses too.

When/where are these courses?

I last thought Phys209 (Phys210) in the Fall (Spring) semester of 2024-2025 academic year.

What are the course policies?

General:

• You accept by taking this course that you have read and agreed my policies in soneralbayrak.com/teaching/MyTeachingPolicy.

• The overall score for the course is calculated with the following algorithm:

  

• The grades awarded previous semesters are available along with a statistical analysis of the students’ performance. The link https://soneralbayrak.com/teaching/files/209_20241_infographic.png takes you to these statistics of the course Phys209 for the $1$st semester of the academic year 2024-2025. You can access other statistics by changing 209 and 20241 appropriately.

Course Material:

• There are some notes that I’ve been updating irregularly, you may check them here. For the source, see the files in github.
• The textbook for Phys209 is “Elementary Differential Equations and Boundary Value Problems” by Boyce and Diprima ($10^{\text{th}}$ edition). The textbooks for Phys210 are “Mathematical Methods in the Physical Sciences” by Boas ($3^{\text{rd}}$ edition) and “Advanced Calculus for Applications” (Hildebrand).
• I list suggested problems from these sources here: feel free to check them!

• For self-study, you may check out the previous courseload although the similarity of the exams among different semesters is not necessarily high:

  1.

  Past exams: The link https://soneralbayrak.com/teaching/files/210_20242_exam3.pdf takes you to the $3$rd exam in the $2$nd semester of the academic year 2024-2025 for the course Phys210. You can access other exams by changing 210, exam3 and 20242 appropriately.

  2.

  Past homeworks: The link https://soneralbayrak.com/teaching/files/209_20231_hw12.pdf takes you to the $12$th homework in the $1$st semester of the academic year 2023-2024 for the course Phys209. You can access other homeworks by changing 209, exam2 and 20241 appropriately.

What will you learn in these courses?

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(1) Definition and Classification of Differential Equations: ordinary vs partial, linear vs non-linear, homogeneous vs non-homogeneous, $1^{\text{st}}$ order vs $2^{\text{nd}}$ order vs $3^{\text{rd}}$ order vs $\dots <!--\; FUNCTION\; APPLICATION\; -->$

(2) Linear equations with constant coefficients: Their relevance in physics and engineering, principle of superposition, characteristic equation, repeated roots, formal solution for both homogeneous and nonhomogeneous cases, initial and boundary value problems, the impulse response, the Laplace transform, convolution

(3) Linear equations with functional coefficients:

1.

Homogeneous solution: Their relevance in physics and engineering, known soluble cases, reduction of order, linear independence and Wronskian, series solutions (Frobenius Method), Fourier transforms and Fourier series

2.

Particular solution: Method of undetermined coefficients, method of variation of parameters

(4) Systems of first order linear differential equations: Conversion of arbitrary linear ordinary differential equations to systems of first order linear differential equations, analysis of coupled systems via matrix differential equations, phase space, formal solutions (Volterra integral equation, Dyson series)

(5) Eigensystems and Sturm-Liouville theory: Matrix diagonalization, normal modes, eigenvalues and eigenfunctions, Sturm-Liouville type differential equations and their solutions

(6) Beyond linear ordinary differential equations:

1.

Non-linear differential equations: Known soluble cases (separable, exact, Bernoulli)

2.

Partial differential equations: Separation of variables, Laplace equation, physical examples

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(1) Functions of a complex variable: analysis of complex numbers as a two dimensional real vector space (complex plane, geometric interpretations, polar form), algebraic properties of complex numbers, series expansions, analyticity of complex functions, complex differentiation & integration

(2) Complex Analysis: Riemann sphere, Riemann surfaces, inner product between complex functions, Fourier analysis, analytic continuation

(3) Definition and Fundamentals of Vector Spaces: basics of linear algebra and linear spaces, norms and inner product spaces, tensor algebras over fields

(4) Curvilinear Coordinate Transformations: coordinate charts, common orthogonal curvilinear coordinates (polar, cylindrical, spherical)

(5) Differentiation in Vector calculus: vector fields in Euclidean spaces, differentiation of vector fields (grad, div, curl, Laplacian), Helmholtz decomposition of vector fields

(6) Integration in Vector calculus: curves and line integrals, Frenet–Serret formulas, integral theorems (gradient theorem, divergence theorem, curl theorem, and Green’s theorem)

What is the course calendar?