Math 303: Applied Mathematics

Spring 2020

Topics Covered in Lectures

Lecture Number	Date	Content	Corresponding Reading material
1	Jan. 28	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	Jan. 30	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, logarithm of a complex number	Pages 94-104 of textbook
3	Feb. 04	Branches of f(z)=z^1/nand the construction of the related Riemann surface, properties of the exponential function e^z, zeros of sin(z) and cos(z), definition of tan(z) and cot(z), the relationship between trigonometric and hyperbolic functions, the logarithm and inverse trigonometric and inverse hyperbolic functions; Complex sequences and series and their convergence	Pages 97-100 & 105-106 of textbook
4	Feb.06	Complex power series and their convergence, special functions; Review of real-valued functions of a several real variables: limit, continuity, directional and partial derivatives, differentiable functions	Pages 151-153 of textbook
5	Feb. 11	Taylor series for a function of a single real variable and the real analytic functions, Taylor series for functions of several real variables, chain rule and coordinate transformations	Pages 157-162 of textbook
6	Feb. 13	The minimum, maximum, and stationary points of a function of a single real variable, the second derivative test for function of a single variable, Generalization to functions of more than one variable, eigenvalue problem for the Hessian, the second derivative test for functions of two variables, extension to functions of more than two variables	Pages 162-167 of textbook
Quiz 1	Feb 14
7	Feb. 18	Second derivative test for functions of n 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
8	Feb. 20	An application of the method of Lagrange multipliers, 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	See Supplementary Material: Pages 181-186, Section 10 of Chapter 4 of Boas's Mathematical Methods in the Physical Sciences
9	Feb. 25	Vector algebra, dot and cross products, Kronecker delta symbol and its properties, Levi Civita epsilon symbol, its properties and applications; Vector fields and their partial derivatives	-
10	Feb. 27	Divergence, curl, and Laplacian, various identities for the divergence and curl of vector fields, differential one-forms, differential of a scalar function, exact differentials	Pages 155-156 & 367-369 of textbook
11	Mar. 03	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 into finitely many rectangles	Pages 377-389 of textbook
12	Mar. 05	Theorem: Let F: R^2→ R^2 be a differentiable force field with components F1 and F2, and ω:=F.dx. Then the following are equivalent: 1) ω is exact; 2) ∂1F2=∂2F1; 3) F is conservative. Divergence theorem in 2D, Divergence theorem in 3D.	Pages 401-402 of textbook
Quiz 2	Mar. 06
13	Mar. 10	Continuity equation and conservation laws, Stokes’ theorem and its topological implications	Pages 404 & 406-408 of textbook
14	Mar. 12	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, 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 824-832 of textbook
15	Mar. 31	Constructing a differentiable function f: C→ C whose real part is a given solution of the Laplace equation, Cauchy-Riemann conditions for the real and imaginary parts of the derivative of the function, differentiable and entire functions, holomorphic functions, review of complex sequences, series, their convergence, and complex power series, analyticity of holomorphic functions, curves and contours in complex plane, integral of a function f: C→ C along a curve, Cauchy’s theorem with proof (using Green’s theorem), Cauchy’s integral formula for a holomorphic function and its derivatives, deformation property of contour integrals	Pages 845-853 of textbook & Pages 592-595 of the handout: Complex-Integration
16	Apr. 02	Laurent Series, poles and essential singularities of complex-valued functions, Laurent series expansions of z/(z^2-2), Residue theorem	Pages 853-861 of textbook & Pages 596-597 of the handout: Complex-Integration
17	Apr. 07	Examples of poles and essential singularities, Extension of the Residue theorem for contours including more than one singularities of the integrand, finding the residue of a pole	Pages 858-861 of textbook & Pages 598-601 of the handout: Complex-Integration
18	Apr. 09	Zeros of holomorphic functions, 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
19	Apr. 14	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	Apr. 16	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	Apr. 21	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, characterization theorems for equality of generalized functions	Pages 103-109 of the handout: Dirac Delta Function
22	Apr. 28	Generalized functions as certain linear functionals, Properties of the Dirac delta function: Relation to evaluation map, evenness and scaling property; derivative of a generalized function, delta function as the derivative of a step function, derivatives of the delta function, delta function evaluated at F(x) for a differentiable function with finitely many zeros that are all simple	Pages 111-112 of the handout: Dirac Delta Function
23	Apr. 30	Series representation of the delta function and the complex Fourier series, the real Fourier series and the Dirichlet’s theorem (without proof)	Pages 415-421 & 424-425 of textbook
24	May 05	Integral representation of the Dirac delta function, the Fourier transform, and its inverse, Parseval’s identity, properties of the Fourier transform, application of Fourier transform in solving linear ODEs	Pages 433-437 of textbook
25	May 07	Some other properties of the Fourier transform, functions whose Fourier transform is a generalized function, convolution formula and its application in solving linear ODEs, solution of the equation of motion for a damped forced oscillator using Fourier transformation, resonance phenomenon	Pages 442-448 of textbook
26	May 12	Generalized functions of several real variables and their equality, Dirac delta function in n dimensions, 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	-
27	May 14	A review of linear algebra: Complex vectors spaces, subspaces, span, linear independence, basis and basis expansion, linear operators, dual space, dual basis, isomorphisms, basis transformations	-
28	May 21	Inner product and complex inner-product spaces, orthogonality, unit vectors, orthonormal bases, unitary operators, space of functions f:{1,2,…,n}-> C and its continuum analog (the space of square-integrable functions), Fourier transform as a basis transformation	-

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