Math 303: Applied Mathematics

Spring 2019

Topics Covered in Lectures

Lecture Number	Date	Content	Corresponding Reading material
1	Feb. 04	Construction of complex numbers, Euler’s formula, modulus and argument of a complex number and its polar representation, U(1) and the rotations in complex plane	Pages 82-94 of textbook
2	Feb.06	Properties and geometric meaning of complex-conjugation, reflections in complex plane, definition and basic properties of the exponential of a complex number, complex polynomials and complex rational functions, roots of complex numbers, trigonometric and hyperbolic functions of a complex variable	Pages 94-99 & 102-104 of textbook
3	Feb. 11	Riemann sheets f(z)=z^1/n with n being a positive integer, logarithm of a complex number, inverse trigonometric and inverse hyperbolic functions of a complex variable; Review of calculus of several variables: Limit and continuity	Pages 97-101 & 105-106 of textbook
4	Feb. 13	Review of calculus of several variables: Directional and partial derivatives, differentiable functions; Taylor series for a function of a single real variable and the real analytic functions, Taylor series for functions of several real variables, differential of a function of several real variables, differential one-forms and their exactness	Pages 131-141 & 151-156 of textbook
5	Feb. 18	Chain rule and coordinate transformations, The stationary, minimum, maximum, and saddle points; the basic idea for the second-derivative test for a real-valued function of two independent variables.	Pages 157-165 of textbook
6	Feb. 20	The second-derivative test for a real-valued function of two independent variables; Finding stationary points of a real-valued function of several variables in the presence of constraints: The method of Lagrange multipliers	Pages 165-173 of textbook, and Sections 8 and 9 of Chapter 4 of Boas's Mathematical Methods in the Physical Sciences
Quiz 1	Feb. 22
7	Feb. 25	Boundary and interior of a subset of the Euclidean space R^n, bounded subsets of R^n, global minima and maxima of a function on a subset of R^n; Vector algebra, dot and cross products, Kronecker delta symbol and its properties	See Supplementary Material: Pages 181-186, Section 10 of Chapter 4 of Boas's Mathematical Methods in the Physical Sciences
8	Feb. 27	The Levi Civita epsilon symbol, its properties and applications; Vector fields and their partial derivatives, divergence, curl, and the Laplacian; Various identities for the divergence and curl of vector field.	Pages 367-369 of textbook
9	Mar. 04	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 its proof for domains that can be dissected in to finitely many rectangles, 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).	Pages 377-389 of textbook
10	Mar. 06	Divergence theorem in plane (R^2) and space (R^3), Continuity equation and conservation laws.	Pages 401-402 & 404 of textbook
11	Mar. 08	Coordinate-independent expression of Green’s theorem, Stokes’ theorem and its topological implications, 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 space (R^3); Complex-valued function of a single complex variable: Correspondence to vector fields in two dimensions, limit and continuity	Pages 406-407 of textbook
Quiz 2	Mar. 15
12	Mar. 18	Definition of a differentiable complex-valued function at a point, Examples of differentiable and non-differentiable complex-valued functions, Cauchy-Riemann conditions, implication for solving the Laplace equation in 2 dimensions	Pages 825-830 of textbook
13	Mar. 20	Verification of the Cauchy-Riemann conditions for e^az and ln(z) and computing their derivatives, Cauchy-Riemann conditions for the real and imaginary parts of the derivative of the function, holomorphic and entire functions, sequences and series of complex numbers and their convergence, complex power series, the analyticity of the holomorphic functions (without proof); Contour integrals, Cauchy’s theorem with proof	Pages 830-832 & 845-849 of textbook
14	Mar. 22	Cauchy’s integral formula, integral formula for the derivatives of a holomorphic function, Laurent Series, poles and essential singularities of complex-valued functions	Pages 849-858 of textbook & Pages 592-596 of the handout: Complex-Integration
15	Mar. 25	Laurent Series for z/(z^2-2) for 0 < |z- and |z- , Residue theorem and its applications, finding the residue at a simple pole, finding the order of a multiple pole	Pages 858-861 of textbook & Pages 596-597 of the handout: Complex-Integration
16	Mar. 27	Determination of the residue of a function at a pole, examples, zeros of holomorphic functions	Pages 598-601 of the handout: Complex-Integration
Midterm Exam
17	Apr. 01	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
18	Apr. 03	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
Spring Break
19	Apr. 15	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
20	Apr. 17	Mass and charge distribution for a point particle, definition of the Dirac Delta function using a simple representative sequence, Schwartz-class functions, equivalent sequences of functions and generalized functions, generalized functions as certain linear functionals	Pages 103-108 of the handout: Dirac Delta Function
21	Apr 24	A characterization theorem for equality of generalized functions, delta function is even, scaling property of the delta function, derivative of a generalized function, the derivative of the step function is the delta function.	Pages 108-109 & 111-112 of the handout: Dirac Delta Function
Quiz 3	Apr 26
22	Apr. 29	Derivatives of the delta function, 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
23	May 03	Integral representation of the Dirac delta function, the Fourier transform, and its inverse, Parseval’s identity	Pages 433-437 of textbook
24	May 06	Fourier transform of the derivatives of a function, application of Fourier transform in solving a linear second order ODE	Pages 442-445 of textbook
25	May 08	Convolution formula and its application in obtaining a particular solution for arbitrary linear non-homogenous ODEs with constant coefficients (computing Green’s functions); A review of linear algebra: Complex vectors spaces, subspaces, span, linear independence, basis and basis expansion, dimension, inner product and complex inner-product spaces, orthogonality, unit vectors, orthonormal bases	Pages 446-448 & 241-245 of textbook
26	May 13	Examples of complex inner-product spaces: Euclidean inner product on C^n, space of n x n complex matrices with inner product: <A,B>=tr(A^+B), space of complex-valued polynomial p:[0,1]-> C with L^2-inner product; linear operators, dual space of a vector space and the dual basis, Riesz lemma, space of functions f:{1,2,…,n}-> C with L^2-inner product and its continuum analog (the space of square-integrable functions); Fourier transform as a basis transformation	-
27	May 15	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, integral representation of delta function of n real variables, the n-dimensional Fourier and inverse Fourier transform	-
Quiz 4	May 17

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