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)=z1/n and 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 ℝn, the
boundary points and boundary of a subset of ℝn |
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 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;
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 |
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: 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: 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.