Announcements

Class No	Date	Content	Corresponding Reading material
1	Sep. 27	Construction of complex numbers, Euler’s formula, polar representation, complex-conjugation, reflections and rotations in complex plane	82-95
2	Sep. 29	Integer powers and roots of complex numbers, complex sequences and series, complex power series and their convergence, trigonometric and hyperbolic functions, logarithm of a complex variable	95-100
3	Oct. 04	Complex powers of a complex number, inverse trigonometric and inverse hyperbolic functions of a complex variable; Relation between inverse hyperbolic functions and the logarithm; Review of calculus of several variable: Differential of a function, Directional and partial derivatives.	102-108, 151- 153,
4	Oct. 06	Taylor series expansion for functions of one or two real variables, Exact differentials, Chain rule.	153-162
5	Oct. 11	Taylor series expansion and the linear and quadratic approximation. The stationary, minimum, maximum, and saddle points; the second derivative test for a real-valued function of several variables. Examples.	162-167
6	Oct. 13	Finding stationary points of a real-valued function of several variables in the presence of constraints: The method of Lagrange multipliers; examples; Global minimum and maximum points.	167-173
Quiz 1
7	Oct. 20	Vector algebra, dot and cross products, Kronecker delta and Levi Civita epsilon symbols and their properties.	212-226
8	Oct. 22	Vector fields and their partial derivatives, the gradient of a scalar field revisited, divergence, curl, and the Laplacian; Various identities for the divergence and curl of vector fields.	347-357
9	Oct. 25	Particle dynamics in Newtonian mechanics, work done by a force, Conservative forces and path independence of their work, Green's Theorem.	377-382, 384
Quiz 2	Oct. 27
10	Nov. 01	Sketh of a proof of Green’s theorem in plane, Proof of the equaivalence of the exactness of the differential 1-form for the infinitesimal work done by a force and the condition that the force is conservative in plane (R^2). Statement of the generalization of this theorem for R^n, Divergence theorem in plane with proof.	384-389
11	Nov. 03	Divergence theorem R^3 with a sketch of proof, continuity equation, and conservation laws	389-405
Midterm Exam 1	Nov. 08
12	Nov. 10	Continuity equation implies a conservation law; Stokes theorem with a sketch of proof, application	405-409
Kurban Bayramı	Nov. 15-19
13	Nov. 22	Dirac Delta function: Generalized functions as equivalence classes of certain sequences of functions	439-442, and the handout: Dirac Delta Function
14	Nov. 24	Test functions, sequences of functions defining generalized functions, definition of the Dirac Delta function using a simple representative sequence, Delta function defined on R^n, Derivation of the Laplacian of 1/r in three dimension	Pages 100-108 of the handout: Dirac Delta Function
15	Nov. 29	Delta function is even, scaling property of the delta function, derivative of a generalized function, derivative of the step function as a generalized function, properties of the derivative of the delta function	Dirac Delta Function
16	Dec. 01	A series representation of the delta function, complex and real Fourier series, Dirichlet’s theorem on the existence of Fourier series expansion of periodic functions (without proof)	415-419
Quiz 3	Dec. 03
17	Dec. 06	Graphical demonstration of the convergence of Fourier series for a step function, differentation of Fourier series, Fourier series for arbitrary intervals, Parseval’s identity for Fourier series	419-432
18	Dec. 08	Fourier transform and inverse Fourier transform, intregral transforms and their linearity, Fourier transform of the derivatives of a function, application of Fourier transform in solving linear ODEs	433-437, 442-444
19	Dec. 13	Derivation of the Green's function for a general constant coefficient 2nd order linear ODE using Fourier transform; Functions of a complex variable: Continuity and differentiability, analytic and entire functions.	824-827
20	Dec. 15	Cauchy-Riemann conditions (proof of necessity and sufficiency for analyticity of the function, examples of analytic functions (z^2, sin z, ln z), implication for solving the Laplace eqn in 2-dim, Cauchy-Riemann conditions for the derivative of an analytic function.	827-835
21	Dec. 20	Review and PS	-
Midterm Exam 2	Dec. 22
22	Dec. 27	Contour integrals, Cauchy’s theorem (proof using Green’s thm in plane), Cauchy’s integral formula,	845-853
23	Dec. 29	Laurent Series, Poles and essential singularities, Residue theorem, determination of the order and the residue of a function at a pole, examples	853-860
24	Jan. 03	Application of the residue theorem in evaluating angular and improper real integrals	861-862
25	Jan. 05	Application of the residue theorem in evaluating various improper real integrals	862-867
Quiz 4	Jan. 07

Note: The pages from the textbook listed above may not include some of the subjects covered in the lectures.