Math 303, Fall 2017

Topics Covered in Each Lecture

Lecture Number	Date	Content	Corresponding Reading material
1	Sep.  18	Construction of complex numbers, Euler’s formula, polar representation, U(1) and the rotations in complex plane	Pages 82-94 of textbook
2	Sep.  20	Complex-conjugation, reflections in complex plane, integer powers and roots of complex numbers, Riemann sheets	Pages 94-99 of textbook
3	Sep.   25	Complex sequences and series, absolute convergence, complex power series and their convergence, exponential and logarithm of a complex number, trigonometric functions of a complex variable	Pages 92-93 & 99-101 of textbook
4	Sep.  27	Hyperbolic, inverse trigonometric, and inverse hyperbolic functions of a complex variable; Review of calculus of several variable: Limit and continuity for a function of several variables, the directional and partial derivatives, differentiable functions; Taylor series for a function of a single real variable and the real analytic functions	Pages 102-106, 131-141, 151-152 of textbook
5	Oct.  02	Taylor series for functions of several real variables, Chain rule and coordinate transformations	Pages 153-162 of textbook
6	Oct.  04	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.	Pages 162-167 of textbook
Quiz 1	Oct.  06
7	Oct.  09	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	Pages 167-173 of textbook, and Sections 8, 9, and particularly 10 of Chapter 4 of Boas's Mathematical Methods in the Physical Sciences
8	Oct.  11	Vector algebra, dot and cross products, Kronecker delta and Levi Civita epsilon symbols, their properties and applications	Pages 213-226 & 941-942 of textbook
9	Oct. 16	Vector fields and their partial derivatives, divergence, curl, and the Laplacian; Various identities for the divergence and curl of vector field; Particle dynamics in Newtonian mechanics, work done by a force, Conservative forces and path independence of their work, statement of the Green's Theorem in plane and the basic idea for its proof for domains that are union of finitely many rectangles that may only intersect only at one side.	Pages 367-369. 377-382 & 384 of textbook
10	Oct. 18	Proof of Green’s theorem in plane for a rectangular domain and its generalization for domains that are union of finitely many rectangles with possible intersection at one side, 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 and proof of the divergence theorem in plane. Statement of the divergence theorem in space (R^3).	Pages 384-389 and 401-402 of textbook
11	Oct. 23	Divergence theorem R^3 with a sketch of proof, continuity equation and conservation laws, statement of Stokes' theorem	Pages 401-406 of textbook
12	Oct. 25	Idea of a proof for Stokes' theorem, applications of Stokes’ theorem; Calculus of functions of a single complex variable: Correspondence with real vector-valued functions in plane, limit and continuity for complex-valued functions, the definition of a differentiable complex-valued function at a point, the comparison with the directional derivative of the corresponding real vector-valued function	Pages 406-407 & 824-825 of textbook
Quiz 2	Oct.  27
13	Oct. 30	Examples of differentiable complex-valued functions, derivative of powers and exponential of z, Examples of non-differentiable complex-valued functions, Cauchy-Riemann conditions (proof of necessity for the differentiability of the function), implication for solving the Laplace equation in 2 dimensions, Cauchy-Riemann conditions for the derivative of the function.	Pages 825-830 of textbook
14	Nov. 01	Verification of the Cauchy-Riemann conditions for e^z and ln(z) and computing their derivatives, Contour integrals, Cauchy’s theorem with proof, Cauchy’s integral formula with derivation, integral formula for the derivatives of a holomorphic function.	Pages 845-853 of textbook
15	Nov. 06	Laurent Series, poles and essential singularities of complex-valued functions	Pages 853-858 of textbook & Pages 592-596 of the handout: Complex-Integration
16	Nov. 08	Residue theorem and its applications, finding the residue at a simple pole	Pages 858-861 of textbook & Pages 596-597 of the handout: Complex-Integration
17	Nov. 13	Determination of the order and the residue of a function at a pole, examples, zeros of holomorphic functions	Pages 598-601 of the handout: Complex-Integration
18	Nov. 15	Singularities of ratios of holomorphic functions, application of the residue theorem in evaluating an angular integral	Pages 861-862 of textbook & Pages 601-603 of the handout: Complex-Integration
Exam 1	Nov. 18
19	Nov. 20	Application of the residue theorem in evaluating improper real integrals with integrand not having a branch cut.	Pages 862-865 of textbook & Pages 603-607 of the handout: Complex-Integration
20	Nov. 22	Application of the residue theorem in evaluating improper real integrals using a contour integral with noncircular contour and in the presence of a branch cut	Pages 865-867 of textbook & Pages 607-609 of the handout: Complex-Integration
21	Nov. 27	Mass and charge distribution for a point particle, definition of the Dirac Delta function using a simple representative sequence, sequences of functions defining generalized functions, a characterization theorem for equality of generalized functions	Pages 103-108 of the handout: Dirac Delta Function
22	Nov. 29	Delta function is even, scaling property of the delta function, derivative of the step function as a generalized function, delta function as the derivative of the Heaviside step function, derivatives of the delta function, delta function evaluated at a function with simple zeros	Pages 108-114 of the handout: Dirac Delta Function
Quiz 3	Dec. 01
23	Dec. 04	Integral and series representations of the delta function, complex and real Fourier series, Dirichlet’s theorem on the existence and convergence of Fourier series	Pages 415-421 & 424-425 of textbook
24	Dec. 06	Integral representation of the Dirac delta function, the Fourier transform, and its inverse.	Pages 433-437 of textbook
25	Dec. 11	Fourier transform of the derivatives of a function, application of Fourier transform in solving linear ODEs; A review of linear algebra: Complex vectors spaces, subspaces, span, linear independence, basis, and dimension	Pages 442-445 of textbook
26	Dec. 13	Complex inner product spaces, unit and orthogonal vectors, orthonormal basis, linear operators, dual space, Dirac’is braket notation, completeness relation associated with orthonormal bases in finite dimensional inner product spaces, a basis for the vector space of linear operators acting in a finite-dimensional inner product space	-
Quiz 4	Dec. 15
27	Dec. 18	Eigenvalue problem for a linear operator, diagonalizable operators acting in a finite-dimensional inner product space, normal operator and their spectral representation, adjoint of a linear operator and Hermitian (self-adjoint) operators acting in a finite-dimensional inner product space, operators (Xf)(x)=x f(x) and (Kf)(x)=-i f’(x) and their spectral representation, Fourier transform as a “basis transformation”	-
28	Dec. 20	Dirac delta function in n dimensions, generalized functions of several real variables and their equality, computation of the Laplacian of 1/r in 3 dimensions, series and integral representation of delta function of n real variables, the n-dimensional Fourier and inverse Fourier transform	-

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