Homework: Summer 2014, Math 435 (Finotti)

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General Guidelines 

Keep this in mind when writing up your homework solutions -- it IS as important for learning math to practice explaining your solutions as it is finding them!  Mathematics is a language.

**On  written homework, quizzes, and exams, solutions need to be well written and explained in order to obtain credit, answers only are not accepted. 

**The purpose of the homework sets are for you to work through the understanding of the material and see what it is you do and do not understand so that you can fine tune your understanding from there -- please keep this viewpoint in mind when working through problems. 

**Educational research shows that studying and learning in peer groups often leads to better comprehension of the material.  Get to know your classmates and form study groups. 

**The other side of the coin is that just as beneficial to your learning is time spent grappling with the material alone.  Always do your final homework writeups on your own, and feel free to reach for help when you need it.  This helps you to know what it is that you really do or don't understand

**If you are suspected of plagerism (think solution manual), you will get a zero for the homework set, and will be reported to University authorities.

*Please look at the following interesting graph of material retention vs. time lapse from first exposure until review (retention curve) and the subsequent tips for textbook reading

Homework Set #

Homework Assigment

      defs 1
 turn in Monday, July 7:
1.1 - define: order of a PDE, a solution of a PDE, linear PDE,  homogeneous/inhomogeneous PDE,
1.2 - define:  directional derivative of a function f(x,y), level curve of a function f(x,y)
      defs 2
 turn in Tuesday, July 8:
1.3 - define the transport equation, the wave equation, the diffusion equation (aka the heat eqn), the Laplace equation
      defs 3
 turn in Wednesday July 9:
1.4 - define: initial condition, boundary condition (b.c.), Dirichlet b.c., Neumann b.c., Robin b.c., homogeneous b.c.
      defs 4
 turn in Thursday July 10:
1.5 - define:  well-posed initial/boundary value problem
1.6 - define:  elliptic PDE, hyperbolic PDE, parabolic PDE
1
  Homework 1 (solutions) - up to and including number 6 can be finished after class on Monday July 7... we will cover how to solve problems like #7 in class on Tuesday July 8.
2
  Homework 2 (solutions) - can be finished after class on  July 10, 2014
defs 5
 turn in Friday July 11:
2.2 - define: principle of causality, domain of influence of (x_0, 0), domain of dependence of (x, t)
      Exam 1 Past exam 1
defs 6
 turn in Wednesday July 15:
2.3 - write the statement of:  Maximum Principle, Strong maximum principle, the comparison principle for the diffusion equation (see problem 6 at the end of 2.3)
2.4 - define the general solution to the diffusion equation on the real line with Dirichlet initial condition, and the error function
3
  Homework 3 (solutions) - can be finished after class on Wednesday July 16!
 defs 7
 turn in Wednesday July 23:
5.1 - define:  fourier sine series and it's coefficients, fourier cosine series and it's coeffs, and full fourier series and it's coeffs
defs 8
 turn in Thursday July 24:
5.3 - define:   symmetric b.c.
         write:  theorem 1, theorem 3
5.4 - define:  pointwise convergence of a series to f(x), uniform convergence of a series to f(x)
         write:  Theorem 2, Theorem 4
4
  Homework 4 (solutions, and a problem like #11 - solution)  (can be finished after class on July 24!)
5
  Homework 5 (solutions)  (SKIP Number 4!)  Can do 1-3 after class on July 24, and #5 after class on July 25.
defs 9
  turn in Wednesday July 31:
6.1 - define: harmonic function, Laplace's equation, the Laplacian, Poisson's equation, rigid motions in 2D, rigid motions in 3D
defs 10
 turn in Thursday July 31:
6.3 - define: Poisson's formula, Theorem 1
defs 11
 turn in Monday August 4:
7.1 -  Green's 1st ID, Mean Value Property in 3D, Max principle in 3D, statement of Dirichlet's principle in 3D
7.2 - Green's 2nd ID, Representation formula
7.3 -  Def of Green's function, theorem 1, theorem 2
6
 Homework 6 (solutions)
Exam 4
 Know: 
- Separation of variables for Laplace's equation on a rectangle and on a disk or annulus (both 2D)
- How to use Poisson's formula to get the Mean Value property in 2D for harmonic functions on disks
- Using Green's 1st ID to get the Mean Value Property in 3D
- Proof of Dirichlet's principle
- Proof of unique solution of the Dirichlet problem  via max/min principle (in 2 or 3D)
- Proof of uniqueness up to a constant for solutions of the Neumann problem
- Proof of theorem 7.3.1
final
exam!
 We'll have roughly one problem over topics covered in each exams material. One problem from material tested over in Exam 1, one from Exam 2 material, etc.