Math 303, Fall 2008
Lecture Contents
|
No |
Date |
Content |
Pages from the textbook or Handouts |
|
|
L1 |
Tue. |
Sep.16 |
Construction of complex numbers, Euler’s formula, polar representation, complex-conjugation, reflections and rotations in complex plane | 82-95 |
|
L2 |
Thu. |
Sep.18 |
Integer powers and roots of complex numbers, complex sequences and series, complex power series and their convergence, trigonometric and hyperbolic functions, logarithm and complex powers of a complex number, inverse trigonometric and inverse hyperbolic functions of a complex variable | 95-105 |
|
L3 |
Tue. |
Sep.23 |
Relation between inverse hyperbolic functions and the logarithm; Review of calculus of several variable: Differential of a function, Directional and partial derivatives, Taylor series expansion for functions of one or two real variables | 105, 151- 153, 160-162 |
|
L4,H1 |
Thu. |
Sep.25 |
Implicit differentiation, Exact Differentials, chain rule | 155-158 |
| Q1 |
Fri. |
Sep.26 | ||
| Holiday Week | ||||
| L5 | Tue. | Oct.07 | Extremum points of real-valued functions, second derivative test and its generalization to functions of two real variables | 162-165 |
| L6 | Thu. | Oct.09 | Extremum points of functions of n real variables, Extremization in presence of constraints, method of Lagrange multipliers | 165-174 |
| L7 | Tue. | Oct.14 | Applications of the method Lagrange multipliers to two concrete problems, boundary (global) stationary points | Handout 1 |
| L8,H2 | Thu. | Oct.16 | Space vectors, scalar and cross products, Kronecker delta and Levi Civita epsilon symbols, their properties and applications. |
212-226 |
| Q2 | Fri. | Oct.17 | ||
| L9 | Tue. | Oct.21 | Vector calculus, gradiaent, divergence, curl, Laplacian, normal vector field to a smooth surface, particle dynamics in Newtonian mechanics, work done by a force | 334-338, 347-354, 356-357 |
| L10 | Thu. | Oct.23 | Conservative forces and path independence of their work, Green's Theorem. | 377-382, 384-389 |
| E1 | Mon. | Oct.27 | ||
| Tue. | Oct.28 | Holiday | ||
| L11 | Thu. | Oct.30 | 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. | 384-389 |
| L12 |
Tue. |
Nov.04 | Divergence theorem in plane with proof, Divergence theorem R^3 with a sketch of proof, continuity equation, and conservation laws | 389-405 |
| L13,H3 |
Thu. |
Nov.06 | Stokes theorem, Dirac Delta function: Generalized functions as equivalence classes of certain sequences of functions. | 405-409 |
| Q3 |
Fri. |
Nov.07 | ||
| L14 |
Tue. |
Nov.11 | Delta function defined on R^n, Properties of the Dirac Delta function: Laplacian of 1/r | 439-442, and Handout 6 |
| L15 |
Thu. |
Nov.13 | Delta function is even, scaling property of the delta function, derivative of a generalized function, an integral representation of the delta function | Handout 6 |
| L16 |
Tue. |
Nov.18 | 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), an example: step function | 415-432 |
| L17,H4 |
Thu. |
Nov.20 | 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; Fourier transform and inverse Fourier transform, intregral transforms and their linearity | 433-437, 442-443 |
| Q4 |
Mon. |
Nov.24 | ||
| L18 |
Tue. |
Nov.25 | Fourier transform of the derivatives of a function, application of Fourier transform in solving linear ODEs, Parseval’s identity for Fourier transform, Fourier transform in higher dimensions | 450-453 |
| L19 |
Thu. |
Nov.27 | Review of Linear algebra: Real and complex vector spaces, subspaces, linear independence, span, basis, linear operator, matrix representation of linear operators, invertible operators, examples | 241-250, 282-285 |
| L20 | Tue. | Dec.02 | 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. | |
| L21,H5 | Thu. | Dec.04 | Stationary points of a functional, Euler-Lagrange eqn., Geodesics in Plane, Geodesics in a sphere | 775-781 |
| Q5 | Fri. | Dec.05 | ||
| Dec.09 | Holiday Week | |||
| L22 | Tue. | Dec.16 | Brachistochrone problem, Functional Derivative, Stationary points of a functional of several variable, Least action principle in mechanics. | 781, 788-789 |
| L23 | Thu. | Dec.18 | Calculus of variations in the presence of constraints, Shape of a uniform cable joining two towers; Functions of a single complex variable: Continuous and differentiable functions | 785-786, 824-827 |
| L24 | Tue. | Dec.23 | Analytic and entire functions., 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. | 827-835 |
| L25 | Thu. | Dec.25 | Cauchy-Riemann conditions for the derivative of an analytic function, Contour integrals, Cauchy’s theorem (proof using Green’s thm in plane), Cauchy’s integral formula, Laurent Series | 845-855 |
| E2 | Mon. | Dec.29 | ||
| L26 | Tue | Dec.30 | Poles and essential singularities, Residue theorem, determination of the order and the residue of a function at a pole, examples, zeros of a analytic functions | 837-839, 855-860 |
| L27 |
Tue. |
Jan. 06 | Application of the residue theorem in evaluating angular and improper real integrals | 861-865 |
| L28,H6 |
Thu. |
Jan. 08 | Application of the residue theorem in evaluating other types of improper real integrals | 865-867 |
| Q6 |
Fri |
Jan. 09 | ||
Note: The pages from the textbook listed above may not include some of the subjects covered in the lectures.
Abbreviations: L:= Lecture, H:=Homework, Q:=Quiz, E:=Exam