Math 303, Fall 2013

Topics Covered in Each Lecture

 

 

Lecture Number

Date

Content

Corresponding Reading material

1

Sep. 16

Construction of complex numbers, Euler’s formula, and polar representation Pages 82-89 of textbook

2

Sep. 18

 Complex-conjugation, reflections and rotations in complex plane, integer powers and roots of complex numbers, complex sequences and series, complex power series and their convergence, trigonometric and hyperbolic functions Pages 89-99, 102-105 of textbook

3

Sep. 23

Logarithm of a complex number, 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.

Pages 99-101, 105-108, 151- 153 of textbook

4

Sep. 24

Taylor series expansion for functions of one or two real variables, Exact differentials, Chain rule. Pages 131-141, 153-162 of textbook

5

Sep. 30

The stationary, minimum, maximum, and saddle points; the second derivative test for a real-valued function of two independent variables, an example; the Hessian matrix, its eigenvalue problem, and the second derivative test for real-valued functions of n independent variables.

Pages162-167 of textbook

6

Oct. 02

 Finding stationary points of a real-valued function of several variables in the presence of constraints: The method of Lagrange multipliers; examples  Pages 167-173 of textbook, Sections 8 & 9 of Chapter 4 of Boas's Mathematical Methods in the Physical Sciences

Quiz 1

Oct. 04

   

7

Oct. 07

Global minimum and maximum points, Vector algebra, dot and cross products, Kronecker delta and Levi Civita epsilon symbols Section 10 of Chapter 4 of Boas's Mathematical Methods in the Physical Sciences & Pages 212-226 of textbook

8

Oct. 09

Applications of the Levi Civita symbol, 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. Pages 347-357 of textbook

Kurban Bayramý

Oct.

14-18

    

9

Oct. 21

Particle dynamics in Newtonian mechanics, work done by a force, Conservative forces and path independence of their work, Green's Theorem in plane. Pages 377-382 and 384 of textbook

10

Oct. 23

Proof of Green’s theorem in plane for a rectangular domain and its generalization for other domains, Proof of the equivalence of the exactness of the differential 1-form defined 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. Divergence theorem in space (R^3). Pages 384-389 and 401-402 of textbook

11

Oct. 30

Review of line and surface integrals Pages 377-382 and 389-395 of textbook

12

Nov. 04

Divergence theorem R^3 with a sketch of proof, continuity equation and conservation laws, Stokes' theorem Pages 401-407 of textbook

13

Nov. 06

Applications of the Divergence and Stokes' Theorems, Gauss and Ampere laws, topological invariance aspects Pages 404 and 408 of textbook

Exam 1

Nov. 08

    

14

Nov. 11

Functions of a complex variable: Continuity and differentiability, Cauchy-Riemann conditions (proof of necessity and sufficiency for analyticity of the function), implication for solving the Laplace eqn in 2-dim, Cauchy-Riemann conditions for the derivative of the function, analytic and entire functions, examples Pages 824-830 of textbook

15

Nov. 13

Verification of Cauchy-Riemann conditions for ln(z), Contour integrals, Cauchy’s theorem (proof using Green’s thm in plane), Cauchy’s integral formula, determinations of derivatives of arbitrary order using Cauchy’s integral formula Pages 845-853 of textbook

Quiz 2

Nov. 15

   

16

Nov. 18

 Laurent Series, poles and essential singularities of complex-valued functions Pages 853-858 of textbook & Pages 592-596 of the handout: Complex-Integration

17

Nov. 20

Residue theorem and its applications Pages 858-861 of textbook & Pages 596-597 of the handout: Complex-Integration

18

Nov. 25

Determination of the order and the residue of a function at a pole, examples, application of the residue theorem in evaluating an angular integral Pages 861-862 of textbook & Pages 598-603 of the handout: Complex-Integration

19

Nov. 27

Application of the residue theorem in evaluating improper real integrals with integrand not having a branch cut Pages 862-865 of textbook & Pages 603-607of the handout: Complex-Integration

Quiz 3

Nov. 29

   

20

Dec. 02

Application of the residue theorem in evaluating improper real integrals with integrand having a branch cut; Motivation for Dirac Delta Function, function sequences Pages 865-867 of textbook

21

Dec. 04

Test functions, sequences of functions defining generalized functions, definition of the Dirac Delta function using a simple representative sequence, a characterization theorem for equality of generalized functions Pages 103-108 of the handout: Dirac Delta Function

22

Dec. 09

Delta function is even, scaling property of the delta function, derivative of the step function as a generalized function, a series representation of the delta function and complex Fourier series Pages 108-114 of the handout: Dirac Delta Function

23

Dec. 11

Pre-exam review session: Contour Integration  

Exam 2

Dec. 13

    

24

Dec. 16

Post-exam problem session  

25

Dec. 18

Real Fourier series, Dirichlet’s theorem on the existence of Fourier series expansion of periodic functions (without proof), an example. Pages 415-419 of textbook  

26

Dec. 23

Derivation of the real Fourier series from the complex Fourier series, Graphical demonstration of the convergence of Fourier series for a step function, An integral representation of the Dirac delta function and the Fourier transform and its inverse

Pages 424-425 & 433-437 of textbook

27

Dec. 25

Properties and examples of Fourier and inverse Fourier transform, Fourier transform of the derivatives of a function, application of Fourier transform in solving linear ODEs Pages 442-444 & 451-453 of textbook

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