Math 303: Applied Mathematics

Spring 2021

Topics Covered in Lectures

Lecture Number	Date	Content	Corresponding Reading material
1	Feb. 16	Construction of complex numbers, complex plane, modulus and argument of a complex number and its polar representation, exponential of an imaginary complex number and Euler’s formula	Pages 82-89 & 92-94 of textbook
2	Feb. 18	Basic properties or the real part, imaginary part, and modulus of a complex number, Moivre’s theorem, unimodular complex numbers and rotations, complex conjugation and reflections, complex polynomials and their zeros, fundamental theorem of algebra, rational functions of a complex variable, integer roots of a complex number and their multi-valuedness	Pages 89-92 & 94-98 of textbook
3	Feb. 23	Branches of f(z)=z^1/nand the construction of the related Riemann surface, exponential of a complex number, properties of the exponential function, logarithm of a complex number, complex powers of a complex number	Pages 97-100 of textbook
4	Feb.25	Basic property of the logarithm, trigonometric and hyperbolic functions of a complex variable, inverse trigonometric and inverse hyperbolic functions of a complex variable, complex sequences and series	Pages 105-106 of textbook
5	Mar. 02	Complex power series and their convergence, special functions; Review of real-valued functions of a several real variables: limit, continuity, directional and partial derivatives, gradient	Pages 151-153 of textbook
6	Mar. 04	Differentiability for real-valued functions of several variables, Taylor series for a function of a single real variable and the real analytic functions, Taylor series for functions of several real variables	Pages 157-162 of textbook
7	Mar. 09	Chain rule and coordinate transformations, local minimum, maximum, and stationary points of a function of several real variables, the second derivative test for a function of a single variable	Pages 162-163 of textbook
8	Mar. 11	Local minimum and maximum points of a function of several real variables are stationary points. The second derivative test for a function of several real variables, saddle points	Pages 163-167 of textbook
Midterm Exam 1	Mar. 14
9	Mar. 16	Finding the local minimum and maximum points of a real-valued function of several variables in the presence of constraints: The method of Lagrange multipliers; Open and closed subsets of ℝ^n, the boundary points and boundary of a subset of ℝ^n	Pages 165-173 of textbook, and Sections 8 and 9 of Chapter 4 of Boas's Mathematical Methods in the Physical Sciences
10	Mar. 18	Interior and closure of a subset of ℝ^n, bounded subsets of ℝ^n, existence theorem for the global minimum and maximum value of a function in a closed and bounded subset of ℝ^n; Vector algebra: ℝ^3and its standard basis, matrix representation of elements of ℝ^3 in its standard basis, the dot product and its properties, generalization to ℝ^n: standard basis, standard matrix representation, dot product; unit and orthogonal vectors, projection of a vector along a unit vector	See Supplementary Material: Pages 181-186, Section 10 of Chapter 4 of Boas's Mathematical Methods in the Physical Sciences & Pages 212-222 of textbook
11	Mar. 23	Kronecker delta symbol, cross-product, Levi Civita epsilon symbol, expressing the cross product of two vectors and the determinant of matrices in terms of the the Levi Civita symbol, statement and proof of the identity expressing sums of products of a pair of Levi Civita symbols in terms of the difference of products of pairs of Kronecker delta symbols	Pages 222-226 of textbook
12	Mar. 25	Applications of the Levi Civita symbols in deriving some basic vector identities. Vector Calculus: Vector-valued functions and their limits, directional and partial derivatives; divergence, Laplacian, and curl, the derivation of some identities involving the divergence, curl, and gradient	Pages 334-338 & 347-357 of textbook
13	Mar. 30	Differential of a scalar function, work done my a force and differential (one) forms, exact differentials and conservative forces; Green’s theorem: Statement, the idea of a proof	Pages 153-156, 377-387 of textbook
14	Apr. 01	Proof of Green’s theorem for a rectangular region, 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: Statement and idea of proof	Pages 387-389 & 401-402 of textbook
Spring Break
15	Apr. 13	Divergence theorem in 3D, proof for an infinitesimal parallelopiped, continuity equation in fluid mechanics and conservation of mass, Stokes’ theorem and some of its consequences	Pages 404 & 406-408 of textbook
16	Apr. 15	Calculus of functions of singles complex variable: Limit and continuity, differentiability and Cauchy-Riemann conditions, functions that are differentiable in an open set and Laplace equation in two dimensions	Pages 824-830 of textbook
17	Apr. 20	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	Pages 580-588 of the handout: Complex-Integration
18	Apr. 22	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, Laurent Series	Pages 845-855 of textbook & Pages 588-594 of the handout: Complex-Integration
Midterm Exam 2	Apr. 24
19	Apr. 27	Isolated, removable, and essential singularities, poles and their order, Residue theorem (with contour encircling a single isolated singularity)	Pages 855-859 of textbook & Pages 594-597 of the handout: Complex-Integration
20	Apr. 29	Examples of poles and essential singularities, Residue theorem (with contour encircling several isolated singularities), finding residue of a function at its poles	Pages 859-860 of textbook & Pages 597-601 of the handout: Complex-Integration
21	May 04	Zeros and isolated zeros of a function, zeros of a holomorphic function and their order, isolated singularities of ratios of two holomorphic functions; Application of residue theorem in evaluating angular integrals and improper integrals with no singularities or branch cuts on the real axis	Pages 861-863 of textbook & Pages 601-604 of the handout: Complex-Integration
22	May 06	Application of the residue theorem in evaluating Cauchy principal value of improper real integrals and integrands having a branch cut.	Pages 863-867 of textbook & Pages 604-609 of the handout: Complex-Integration
Bayram Holidays
23	May 18	Mass and charge distribution for a point particle, definition of the Dirac Delta function using a simple representative sequence, smooth functions vanishing outside a finite interval as test functions, equivalent sequences of functions and generalized functions,	Pages 103-109 of the handout: Dirac Delta Function
24	May 20	Evenness and scaling property of the Dirac Delta function; derivative of a generalized function, delta function as the derivative of a step function, delta function evaluated at F(x) for a differentiable function with finitely many zeros that are all simple; Series representations of the the Dirac Delta function and complex and real Fourier series expansions of the test functions	Pages 111-112 of the handout: Dirac Delta Function, and Pages 415-421 & 424-425 of textbook
Midterm Exam 3	Mat 23
25	May 25	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, functions whose Fourier transform is a generalized function	Pages 433-445 & 450-451of textbook
26	May 27	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, 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	Pages 446-449 of textbook

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