Math 303, Fall 2006

Topics Covered in Each Lecture

 

Lecture No

Date

Content

Pages from the textbook

1

Sep.

19

Construction of complex numbers, Euler’s formula, polar representation

82-89, 92-94 

2

 

21

Reflections and rotation in complex plane, integer powers and roots of complex numbers

89-91, 95-98

3

 

26

Complex sequences and series, complex power series and their convergence theorem (without proof), trigonometric and hyperbolic functions, logarithm and complex powers of a complex number

 99-105

 

4

 

28

Inverse trigonometric and inverse hyperbolic functions of a complex variable, Review of calculus of several variable: Differential of a function, Directional and partial derivatives, Taylor series expansion

105-109, 151- 153, 160-162
5 Oct. 03 Exact Differentials, chain rule, coordinate transformations, Extremum points of functions of two real variables

155-160, 162-164
6   05 Extremum points of functions of n real variables, Extremization in presence of constraints, method of Lagrange multipliers

165-174
7   10 Another example of Extremization in presence of constraints, boundary (global) stationary points See Handout 1
8   12 Space Vectors, scalar and cross products, Kronecker delta and Levi Civita epsilon symbols, their properties and applications.

212-226
9   17 Vector calculus, gradiaent, divergence, curl, Laplacian, normal vector field to a smooth surface, particle dynamics in Newtonian mechanics, conservative forces 334-338, 347-354, 356-357 
10   19 Green’s theorem in plane (with sketch of a proof), Divergence 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. 384-389
11   31 Divergence theorem in R^3, continuity equation, Stokes theorem 389-409
12 Nov. 02 Dirac Delta function: Generalized functions as equivalence classes of certain sequences of functions. -
13   09 Delta function defined on R^n, Properties of Dirac Delta function: Laplacian of 1/r, delta is even, scaling property of delta function 439-442, and Chapter 5 of Kuuse-Westwig
14   14 Derivative of a generalized function, integral and series representations of the Dirac delta function, complex and real Fourier series Chapter 5 of Kuuse-Westwig
15   16 Dirichlet’s theorem on the existence of Fourier series expansion of periodic functions (without proof), an example + graphical demonstration, differentation of Fourier series, Fourier series for arbitrary intervals 415-432 
16   21 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, Parseval’s identity, Fourier transform in higher dimensions 433-453
17   23 Review of Linear algebra: Real and complex vector spaces, subspaces, linear independence, span, basis, linear operator, matrix representation of linear operators, invertible operators, examples  
18   28 Expressing the vector spaces R^n and C^n as function spaces. Inner product on C^n, L^2 inner product, Fourier transformation as a basis transformation.  
19   30 Stationary points of a functional, Euler-Lagrange eqn., Geodesics in Plane 775-781
20 Dec. 05 Geodesics in a sphere, Shape of a uniform cable joining two towers, Brachistochrone problem.  
21   07 Stationary points of a functional of several variable, Least action principle in mechanics, Functions of a single complex variable: Continuous and differentiable functions, analytic and entire functions. 782, 788-789, 824-826782
22   12 Cauchy-Riemann conditions (proof of necessity and sufficiency for analyticity of the function), examples of analytic functions (z^2, 1/z, 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
23   14 Contour integrals, Cauchy’s theorem (proof using Green’s thm in plane), Cauchy’s integral formula, Laurent Series 845-855
24   21 Poles and essential singularities, Residue theorem, evaluation of the residue at a simple pole, applications 855-860
25   26 Residue at a multiple pole, application of the residue theorem in evaluating angular integrals 837-839, 861-862
26   28 Application of the residue theorem in evaluating improper real integrals 862-867

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