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 ℝn, the boundary points and boundary of a subset of ℝn, interior and closure of a subset of ℝn, bounded subsets of ℝn, existence theorem for the global minimum and maximum value of a function in a closed and bounded subset of ℝn |
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: ℝ3 and its standard basis, matrix representation of elements of ℝ3 in its standard basis, the dot product and its properties, generalization to ℝn: 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: ℝ2→ ℝ2 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.