• Instructor : Dr. L. Vu-Quoc, 135 NEB, Tel: 392-6227, E-mail: vu-quoc AT ufl.edu
• Classroom: 230 FLG (Section 1298)
• Class time: Tue and Thu, periods 2 + 3 (8:30am - 10:25am)
  MAE course schedule for Spring 2004

Office hours:
Tue, periods 4 + 5 (10:40am - 12:35pm)
Thu, period 4 (10:40am - 11:30am)

Week 15: Thu, 15 Apr 2004
Tue, 20 Apr 04: Exam 3.
Tue, 20 Apr 04: HW13 due for extra credits.

Week 14:
Lectures 27 + 28 / Reading: Thu, 15 Apr 2004

• Sec 10.6: Solving for the coefficients A_n, B_n of the harmonic solution y_n; use of orthogonality condition of cosine and sine to obtain two equations for two unknown coefficients.
• Sec 11.1: Read on your own.
• Sec 11.3: Wave equation, partial differential equation, applications, dimensional analysis for coefficient c as velocity, initial conditions, example with second-order ODE, boundary conditions; solution method: Separation of variables, decouple PDE into two ODEs, search for solution satisfying boundary conditions, general method for linear second-order ODEs, relations for sine and cosine in terms of exponential functions, linear and nonlinear differential operators, superposition of solutions, satisfying initial conditions, application of Fourier series.

Week 13:
Lectures 25 + 26 / Reading:

• Sec 10.4: Fourier series of sum of two functions of same period (cont'd), proof. Half range expansion, proof of formulas for Fourier coefficients for even function expansion.
• Sec 10.6: Oscillations and Vibrations. Application of Fourier series. Circuit example: RLC circuit in series, relationship between drop of potentials and excitation source, potential-current relations for each component, derivation of equation of oscillation in terms of current, second-order ODE. Spring-mass-damper system: Spring, mass, damper in series; free-body diagrams, equation of equilibrium, force-displacement relations for each component, equation of motion, second-order ODE. Comparison of the two governing equations, physical contribution to solution by each term. Numerical example: periodic excitation, Fourier series expansion, superposition of solution for each Fourier harmonic term, frequency and phase, harmonic solution.
• Sec 11.1: Read on your own.

Week 12:
Lectures 23 + 24 / Reading:

• Secs 10.3 - 10.4: Fourier series in compact format, orthogonality of Fourier basis functions, integration method, diagonal Gram matrix, Euler formulas for Fourier coefficients. Non-orthogonal basis functions: Polynomials, example, full Gram matrix, projection of non-periodic functions on non-orthogonal basis of polynomials. Numerical example: Half-wave rectifier; recall Mean Value Theorem, average, DC component; angular frequency, period, frequency. Even functions, odd functions, examples, decomposition of any general function into the sum of an even function (cosine series) and an odd function (sine series), Fourier series expansion of even functions (cosine series) and of odd functions (sine series), proof. Fourier series of sum of two functions of same period.

• First-Known Double Pulsar Opens up New Astrophysics (animations of pulsar motion), University of Manchester, 8 Jan 2004.
• Introduction to pulsar astronomy: Nice short lecture.
• Discoverers of pulsars: Jocelyn Bell and Anthony Hewish. Astronomical injustice.
• How to attract a rock star to study for a PhD degree: Astronomy ("Night life", Nature, 13 Nov 2003)

Week 11:
Lectures 21 + 22 / Reading:

• Secs 10.2 - 10.3: Numerical examples on the computation of Fourier series coefficients (by TAs); Fourier's series, analogy with vectors, 2 methods to compute the components of a given vector with respect to a non-orthonormal basis, examples, method based on inner product, Gram matrix, generalize to functions, orthogonality of Fourier basis functions, Euler's formula for Fourier coefficients.
• Fourier series animations: Convergence, Gibbs's phenomenon

Week 10:
Lectures 19 + 20 / Reading:

• Secs 10.1 - 10.3: Periodic functions, examples, characterization, Fourier basis functions with period 2*pi, Fourier series, functions with period not equal to 2*pi, basis functions, Fourier series, analogy with vectors, computation of components of a given vector with respect to a non-orthonormal basis, example.

Plan: From now until the end of the semester, we'll do
Secs 10.1-10.4, 10.6. (Fourier series)
Secs 11.1-11.5, 11.12. (Partial differential equations)

Spring Break: Mon, 8 Mar 04, to Fri, 12 Mar 04.

Week 9:
Lectures 17 + 18 / Reading:

• Sec 3.0: Eigenvalue problem (cont'd), numerical examples for solving eigenvalue problems (by TA), motivation, solving Ax = b, diagonalization of a matrix, Jordan canonical form, eigenvalue problem in matrix form, transformation of coordinates, uncoupling equations, solution in new coordinates, transform solution back to orginal coordinates, connection of theory to numerical examples.
• Sec 3.1: Solution for systems of first-order differential equations, parallelism with solving A x = b, eigenvalue problem, transformation of coordinates, uncoupling differential equations, solution of uncoupled differential equations, transformation back to original coordinates, connection with numerical example in book.
  Remark: Schrodinger equation in quantum mechanics as a continuous eigenvalue problem, parallelism with discrete (matrix) eigenvalue problems.

Week 8:
Lectures 15 + 16 / Reading:

• Sec 9.7: divergence thm: proof for 2-D case (cont'd), director cosines, relation with dx, dy, and ds (infinitesimal length on boundary), domains with corners.
• Sec 9.9: Stokes's theorem, Green's thm as a particular case, application examples (integration of Ampere's law across a conducting electric wire, magnetic field H as a function of radial variable r; Ref: Ulaby, pp.202-204 ), another interpretation of curl (normal flux, mean value theorem, circulation); read the proof on your own.
• Sec 3.0: Matrix algebra (transpose of product of matrices, inverse of product of matrices, proof), eigenvalue problem, motivation.

Week 7:
Lectures 13 + 14 / Reading:

• Sec 9.5 and 9.6: (cont'd) Parameterization of surfaces for surface integrals.
• Matlab symbolic toolbox: Tutorial 3
• Sec 9.7: divergence thm: Statement, piecewise smooth surfaces (meaning, examples, hat function, spline curve, discontinuities of derivatives), indicial notation, summation convention, 1-D, 2-D, 3-D cases, application example (integration over a sphere, parameterization of a sphere, tangent vectors, normal vector), proof for 2-D case.
• Sec 9.8: Further applications of divergence thm: Derivation of the continuity equation, meaning of flux integral through a closed surface as outflow minus inflow.
• Further motivation: Laws used in semiconductor physics that are like the continuity equation, electron continuity equation, Boltzmann transport equation for carriers, review of Ohm's law for electric current in vector form, coupling between Ohm's law and Fourier's law for heat current (flux).
  Ref: M. Lundstrom, Fundamentals of Carrier Transport (2nd edition), Cambridge U Press, 2000.
• Exam 1: Return and comments.

Week 6:
Lectures 11+12 / Reading:

• Sec 9.2: Criterion for exactness of path-independent line integrals; Stokes's theorem for converse proof.
• Sec 9.3: Read on your own (review of calculus).
• Matlab symbolic toolbox: Tutorial 1
• Sec 9.4: Green's theorem; general statement, application examples, proof.
• Matlab symbolic toolbox: Tutorial 2
• Sec 9.5 and 9.6: Parameterization of surfaces for surface integrals, to be continued next week.

Week 5:
Lectures 9+10 / Reading:

• Sec 9.1: Line integrals (cont'd)
  Work of a force acting on a particle moving on a curve, case of inertia force, kinetic energy.
  Thm 1, p.469, invariance of line integrals under change in parameterization of the curve, smooth function in change of parameterization; proof of Thm.
  Example of functions that are not smooth, functions with discontinuous derivatives, finite element method.
• Sec 9.2: path-independent line integrals
  Example, selection of convenient curve or path, use of different parameterization on the curve (illustration of Thm 1, K, p.469, Sec. 9.1, on change of parameterization on a curve), case of vector-valued function (e.g., force field) derived from a scalar potential, direct evaluation of line integral, integration to find expression of potential, simple way to obtain line-integral result using scalar potential.
  Proof of Thm 1, K, p.471.
  Conservative forces, skiing story as example.

Week 4:
Lectures 7+8 / Reading:

• Sec 8.9: cont'd. Applications:
  Electric field as gradient of electric potential, Ohm's law.
  Fourier's law of heat conduction (gradient of scalar field),
  Hooke's law in elasticity (gradient of displacement-vector field).
• Maxwell's equations: Gauss's law of electric field, Gauss's law of magnetic field, Ampere's law, Faraday's law.
• Sec 8.10: Divergence of vector field;
  Continuity equation in continuum (fluid) mechanics, detailed proof in 1-D then 3-D case, incompressibility condition; meaning of divergence.
  Maxwell's equations: Gauss's laws (for magnetic field), similarity with incompressibility condition of fluids.
  Tensors, orders (0th, 1st, 2nd, 4th), Leibniz formula for divergence of product of a scalar field and a vector field.
  Ref: F. T. Ulaby, Fundamentals of Applied Electromagnetics, Prentice Hall, 2001.
• Sec. 8.11: Curl, rotation, meaning of curl; curl of grad, div of curl and connection with Ampere's law and Faraday's law.
  Gibbs's nabla notation, method to remember formulas in vector calculus, example of Gibbs's nabla notation with divergence of product of a scalar field and a vector field (Leibniz formula), danger of misinterpretation ( Ulaby, p.125).
• Sec 9.1: Line integrals, defining formula, pictorial interpretation of formula, parameterization, example of computation, work of a force along a curve and connection to kinetic energy.

Week 3:
Lecture/Reading:

• Sec 8.6: Acceleration (normal, tangential), Coriolis force.
• Sec 8.8: Read on your own. HW will be assigned.
• Sec 8.9: Motivational story: Hiking in a mountainous area, rate of change of elevation, iso-elevation contours on a map, directional derivative, gradient of scalar fields. Applications:
  Gravitational potential,
  Electric field as gradient of electric potential.
  Ref: F. T. Ulaby, Fundamentals of Applied Electromagnetics, Prentice Hall, 2001.

Week 2:
Lecture/Reading:

• Sec 8.3: Example 5, p.412, a different approach.
• Sec 8.3: proof of Schwarz inequality, triangle inequality, parallelogram equality, generalization to functions.
• Sec 8.4: Scalar and vector fields; vector calculus, derivatives, proof of generalized Leibniz relations for vectors.
• Sec 8.5: Curves, tangent, arc length
• Sec 8.6: Velocity, derivative of constant-length vectors, acceleration (normal, tangential).

Week 1:
Lecture/Reading:

• Course organization.
• Sec. 8.1-8.3: Vector fields, dot product, cross product, scalar and vector triple product, linear independence/dependence of a set of vectors, test of linear independence.
• Sec 6.6: Laplace expansion of determinants, n x n matrices.
• Properties of inner products, generalization of inner-product concept to functions (in preparation for Fourier series).

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Home page: Loc Vu-Quoc E-mail: vu-quoc AT ufl.edu MAE Department