Math 5450   (Dr. J. S. Zheng)


Office: Math-Phys 3306
Phone: (912)478 1338
E-mail: szheng at GeorgiaSouthern dot edu

 


   
    		My favorite equation

The universe is written in the language of mathematics (Galileo)

What is a net-invisible ElectroMagnetic field?

answer:
  • (a) a quantum-scholar?

  • (b) t.b.a.  
    
   


 \[ U_t+ \alpha \cdot \nabla U=\nu \Delta U+g(t,x,U)\]                              Gravitational-wave image Courtesy:Wiki

Mathematical physics with PDE gauge-interface.

Math-Physics lab simulations link


 

Fourier series

\( f(x)=\sum^{\infty}_{k=-\infty} c_k e^{ikx}\)
  \( c_k=\frac{1}{2\pi}\int_0^{2\pi} f(x)e^{-ikx} dx\,. \)




       




Schedule of the class
Class schedule by chapters/sections (tentative)
PDEs from physics; self-adjoint operators (Chapters 1)
Assignment 1 (homework/project)
Wave and diffusion equations, Fourier integrals (Chapter 2)
Separation of variables (Chapter 3)  
Fourier series (Chapter 4)
Assignment 2 (homework/project)
Laplace equation and Poisson equation (Chapter 5)  
Green's function and distribution theory (Chapter 6)  
Assignment 3* (homework/project)  
Maxwell equations for magnetic field (Chapter 7*)  
Euler equation and fluids (Chapter 8*)  
Water wave model (Chapter 9*)  
Tensors calculus and differential forms (Chapter 10*)  
*Numerical methods (Chapters 11*)
Presentation (group project)
*Probabilistic method; statistical mechanics (Chapter 12*)
Final assignment



Homework/Project assignments


Office hours   Monday and Wednesday 2:30-4:00 or by appointment


Important calendar dates


Review   Exam     Date  
Review Exam I     Assigment I
Review Exam II Assignment II
Review Exam III    Assignment III  
Review Final      Final assignment  

Course description

In this newly developed Math-Physics course, students will be introduced to the basic science of physics via mathematical methods and perspectives. Concepts include
  • [Mathematics]: Hilbert space, self-adjoint differential operators, Fourier and Laplace transforms, three type of PDE models arising in classical/quantum physics
  • [Physics]: Newtonain and quantum mechanics-- motion, mass, momentum and energy, harmonic oscillator, Coulomb and delta potential, heat diffusion and temperature, waves (sound, light, radiation), electromagnetics, gravity and relativity. These concepts are elaborated and explored through lecture, reading, supplementary reference, online presentation and/or lab demonstration.


    • Objectives

    Upon successful completion of the course, students will be able to:
  • State definitions, properties and fundamental theory (e.g., inner product space, eigenvalue problem, Fourier series, three typical PDEs
  • Explain and interpret physical mechanisms by solving Math models in an orderly and logical manner.
  • Use vectorial calculus, ODE and PDE to analyze, compute and simulate a priori results that help understand how the physical world works; meanwhile appreciate and gain the mathematical insight.
  • **Time permitting, present a short description about one of the approaches in solving a PDE: analytical, numerical, experimental, or data-driven


    • Quantum leanring, computing, and vectorization  Numerical PDE

  • From Q-bits to large scale, entanglement, probabilistic states

    •   


    Computing Lab