Math 303: Applied Mathematics

Spring 2024

Topics Covered in Lectures

Textbook: K. F. Riley, M. P. Hobson, and S. J. Bence “Mathematical Methods for Physics and Engineering,” 3rd Edition (Cambridge University Press, Cambridge, 2006)

Lecture Number	Date	Content	Corresponding Reading material
1	Feb. 12	Construction of complex numbers, inverse of a nonzero complex number, complex plane, real and imaginary parts of a complex number, modulus and argument of a complex number	Pages 82-91 of textbook
2	Feb. 14	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,	Pages 92-97 of textbook
3	Feb. 19	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, exponential and logarithm of a complex number, complex powers of a complex number	Pages 97-100 of textbook
4	Feb.21	Trigonometric and hyperbolic functions of a complex variable, inverse trigonometric and inverse hyperbolic functions of a complex variable	Pages 102-106 of textbook
5	Feb. 26	Review of classical functions of a complex variable, multi-valuedness of the integer roots of a complex number, branch cuts, Riemann sheets, and Riemann surfaces, the relation between trigonometric and hyperbolic functions and their inverses, complex power series and special functions	Pages 97-106 of textbook
6	Mar. 01	Review of real-valued functions of a several real variables: limit and continuity, 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
7	Mar. 06	Taylor series for functions of several real variables and the Hessian, Chain rule and coordinate transformations	Pages 157-162 of textbook
8	Mar. 08	Local minimum, maximum, and stationary points of a function of several real variables, the second derivative test for functions of one or several real variables	Pages 163-167 of textbook
9	Mar. 11	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)
10	Mar. 13	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
11	Mar. 18	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, expressing the cross product of two vectors and the determinant of matrices in terms of the the Levi Civita symbol	Pages 212-224 of textbook
12	Mar. 20	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, 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 224-226, 334-338 & 347-355 of textbook
Midterm Exam 1	Mar.21
13	Mar. 23	Differential of a scalar function, work done by a force and differential (one) forms, exact differentials and conservative forces; Green’s theorem: Statement	Pages 153-156, 377-387 of textbook
14	Mar. 27	Green’s theorem: Idea of a proof, proof 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	Pages 387-389 & 401 of textbook
15	Apr. 01	Divergence theorem in 3D: Statement, idea of proof, proof for an infinitesimal parallelepiped, Continuity equation in fluid mechanics and conservation of mass, Stokes’ theorem and some of its consequences	Pages 401-402, 404 & 406-407 of textbook
16	Apr. 03	Application of Stokes’ theorem in characterizing conservative forces in 3D; Calculus of functions of a single complex variable: Limit and continuity, derivative and Cauchy-Riemann conditions, relation to solutions of the Laplace equation in 2D.	Pages 824-830 of textbook
Spring Break
17	Apr. 22	Constructing differentiable functions of a complex variable whose real part is a given solution of the Laplace Eq., holomorphic functions, Cauchy-Riemann conditions for the real and imaginary parts of the derivative of the function; Complex sequences, series, and power series, analytic functions of a complex variable	Pages 580-588 of the handout: Complex-Integration
18	Apr. 24	Integration of a function f: C→ C along a curve in complex plane, contours and contour integrals, 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 845-853 of textbook & Pages 588-595 of the handout: Complex-Integration
19	Apr. 29	Isolated, removable, and essential singularities, poles and their order, Residue theorem (with contour encircling a single isolated singularity), examples of poles and essential singularities, Residue theorem (with contour encircling several isolated singularities).	Pages 853-861 of textbook & Pages 595-597 of the handout: Complex-Integration
20	May 06	Finding residue of a function at its poles, Application of residue theorem in evaluating angular integrals	Pages 861-862 of textbook & Pages 598-602 of the handout: Complex-Integration
21	May 08	Improper integral and their principle value, application of residue theorem in evaluating Cauchy principal value of the improper real integrals with no branch cuts on the real axis.	Pages 863-865 of textbook & Pages 604-607 of the handout: Complex-Integration
Midterm Exam 2	May 09
22	May 13	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, Dirac delta function and function sequences representing it	Pages 863-867 of textbook, Pages 604-609 of the handout: Complex-Integration & Pages 102-103 of the handout: Dirac Delta Function
23	May 15	Test functions, equivalent sequences of functions, and generalized functions; The definition of the Dirac delta function, its evenness and scaling property, product of a function and 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 20
25	May 22

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