Math 303: Applied Mathematics

Spring 2022

Topics Covered in Lectures

Lecture Number	Date	Content	Corresponding Reading material
1	Feb. 15	Construction of complex numbers, inverse of a nonzero complex number, complex plane, real and imaginary parts of a complex number, modulus and complex-conjugate of a complex number	Pages 82-91 of textbook
2	Feb. 17	Basic properties of the real part, imaginary part, and modulus of a complex number, polar representation of complex numbers, exponential of an imaginary complex number and Euler’s formula, 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	Pages 92-97 of textbook
3	Feb. 22	Integer roots of a complex number and their multi-valuedness, branches of f(z)=z^1/nand the corresponding Riemann surface, logarithm of a complex number, complex powers of a complex number	Pages 97-100 of textbook
4	Feb.24	Trigonometric and hyperbolic functions of a complex variable, inverse trigonometric and inverse hyperbolic functions of a complex variable; Review of real-valued functions of a several real variables: limit and continuity	Pages 102-106 of textbook
5	Mar. 01	Directional and partial derivatives, gradient, differentiability for real-valued functions of several variables, Taylor series for a function of a single real variable and real analytic functions	Pages 151-153 & 136-141 of textbook
6	Mar. 03	Taylor series for functions of several real variables and the Hessian, Chain rule and coordinate transformations, local minimum, maximum, and stationary points of a function of several real variables	Pages 157-162 of textbook
7	Mar. 08	The second derivative test for functions of one or several real variables, saddle points	Pages 163-167 of textbook
Snow break
8	Mar. 15	Applications of the second derivative test for finding local min. max. and saddle points of functions of several real variables. 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	Pages 167-173 of textbook, and Sections 8 and 9 of Chapter 4 of Boas's Mathematical Methods in the Physical Sciences (See Supplementary Material)
9	Mar. 17	Applications of the method of Lagrange multipliers; Open and closed subsets of ℝ^n, the boundary points and boundary of a subset of ℝ^n, 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	See Supplementary Material: Pages 181-186, Section 10 of Chapter 4 of Boas's Mathematical Methods in the Physical Sciences
10	Mar. 22	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, Kronecker delta symbol, cross-product, Levi Civita epsilon symbol	Pages 212-224 of textbook
11	Mar. 24	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, norm of the cross product of two vectors	Pages 224-226 of textbook
Midterm Exam 1	Mar. 27
12	Mar. 29	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. 31	Differential of a scalar function, work done by a force and differential (one) forms, exact differentials and conservative forces; Green’s theorem: Statement, the idea of a proof, Proof of Green’s theorem for a rectangular region	Pages 153-156, 377-387 of textbook
14	Apr. 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: Statement, idea of proof, proof for an infinitesimal parallelepiped	Pages 387-389 & 401-403 of textbook
15	Apr. 07	Continuity equation in fluid mechanics and conservation of mass, Stokes’ theorem and some of its consequences	Pages 404 & 406-408 of textbook
Spring Break
16	Apr. 19	Calculus of functions of singles complex variable: Limit and continuity, differentiability and Cauchy-Riemann conditions, entire functions, connection to solutions of the Laplace equation in two dimensions	Pages 824-830 of textbook
17	Apr. 21	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	Pages 580-588 of the handout: Complex-Integration
18	Apr. 26	Radius of convergence of power series; Contours and contour integral of a function f: C→ C, 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 588-592 of the handout: Complex-Integration
19	Apr. 28	Laurent Series, isolated, removable, and essential singularities, poles and their order, Residue theorem (with contour encircling a single isolated singularity)	Pages 853-859 of textbook & Pages 592-597 of the handout: Complex-Integration
Bayram Holidays
20	May 10	Examples of poles and essential singularities, Residue theorem (with contour encircling several isolated singularities), finding residue of a function at its poles, zeros of a holomorphic function, isolated singularities of ratios of two holomorphic functions	Pages 859-860 of textbook & Pages 597-601 of the handout: Complex-Integration
21	May 12	Application of residue theorem in evaluating angular integrals and the Cauchy principal value of the improper real integrals with no branch cuts on the real axis.	Pages 863-865 of textbook & Pages 604-606 of the handout: Complex-Integration
22	May 17	Application of residue theorem in evaluating improper real integrals with a branch cut on the real axis; Generalized functions: Mass and charge distribution for a point particle, sequences of functions representing mass density of an arbitrarily short string	Pages 863-867 of textbook, Pages 604-609 of the handout: Complex-Integration & Pages 102-103 of the handout: Dirac Delta Function
Midterm Exam 2	May 21
23	May 24	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; 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	Pages 103-114of the handout: Dirac Delta Function & Pages 439-442 of textbook
24	May 26	Product of a function and the Dirac Delta function; Series representations of the Dirac Delta function and complex and real Fourier series; Integral representation of the Dirac delta function, the Fourier transform, and its inverse, Parseval’s identity	Pages 415-421, 424-425, 433-435 & 450-453 of textbook
25	May 27	Fourier transform of the derivatives of a function, convolution formula for the Fourier transform and its application in solving linear ODEs; 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.