275- Harmonic functions and harmonic conjugates

December 23, 2008

Recall that a function u(x,y) of two variables defined on an open domain D is harmonic iff  u is C^2 (i.e., all four second order derivatives u_{xx},u_{xy},u_{yx},u_{yy} exist and are continuous in D), and u satisfies Laplace equation

u_{xx}+u_{yy}=0.

As mentioned in problem 6 of the Fall 2008 Calculus III final exam, a function v is a harmonic conjugate of u iff v is defined on D, v_x and v_y exist, and the Cauchy-Riemann equations hold:

u_x=v_y and u_y=-v_x.

It follows immediately from the Cauchy-Riemann equations that if v is a harmonic conjugate of a harmonic function u, then v is also C^2, with v_{xx}=-u_{yx}, v_{xy}=-u_{yy}, v_{yx}=u_{xx} and v_{yy}=u_{xy}. It is also immediate that v satisfies Laplace equation because v_{xx}+v_{yy}=-u_{yx}+u_{xy}=0, since continuity guarantees that the mixed partial derivatives commute. Thus v is also harmonic.

In fact, modulo continuity of the second order derivatives, the harmonic functions are precisely the functions that (locally) admit harmonic conjugates.

To see this, assume first that u is C^2 in D and that it admits a harmonic conjugate v. Then u_{xx}=v_{yx} and u_{yy}=-v_{xy} so u_{xx}+u_{yy}=0 and u was harmonic to begin with.

Conversely, assume that u is harmonic in D. Suppose first that D is (connected and) simply connected. I claim that then u admits a harmonic conjugate v in D. To see this, letting {\mathbf F}=(-u_y,u_x), notice that the existence of v is equivalent to the claim that {\mathbf F} is a gradient vector field, since {\mathbf F}=\nabla v iff v is a harmonic conjugate of u. But, since D is simply connected, then {\mathbf F} is a gradient iff it is conservative, i.e., \displaystyle \oint_\gamma {\mathbf F}\cdot d{\mathbf r}=0 for any simple piecewise smooth loop \gamma in D. Fix such a \gamma, and let R denote its interior. Then, by Green’s theorem,

\displaystyle \oint_\gamma {\mathbf F}\cdot d{\mathbf r}=\pm\iint_R u_{xx}+u_{yy}\,dA=0,

where the \pm sign is to be chosen depending on the orientation of \gamma. It follows that {\mathbf F} is indeed conservative and therefore a gradient, so u admits a harmonic conjugate.

Finally, if D is not simply connected, we cannot guarantee that such a v exists in all of D, but the argument above shows that it does in any open (connected) simply connected subset of D, for example, any open ball contained in D.  That we cannot extend this to all of D follows from considering, for example, u(x,y)=\log(x^2+y^2) in D={\mathbb R}^2\setminus\{(0,0)\}. This is a harmonic function but it does not admit a harmonic conjugate in D, since there is no continuous \arctan(y/x) in D. This example can be easily adapted (via a translation) to any non-simply connected D.

I close by remarking that, as mentioned in my previous post on average values of harmonic functions, one can use Green’s theorem to prove that harmonic functions u satisfy the average (or mean) value property, and this property characterizes harmonicity as well, implies that u is actually C^{\infty} (i.e., u admits partial derivatives of all orders, and they are all continuous) and has the additional advantage that it only requires that u is continuous, rather than C^2. Similarly, one can show that the Cauchy-Riemann equations on D suffice to guarantee that u and v are harmonic (and in particular, C^\infty). However, one needs to require that the equations hold everywhere on D. A pointwise requirement would not suffice. But I won’t address this issue here (I mention it in the notes in complex analysis that I hope to post some day).


175, 275 -Final exam

December 13, 2008

 Just a few general remarks:

  • 175: The exam is this Monday, Dec. 15, from 8:00 to 10:00 at the usual place.
  • 275: The exam is this Monday, Dec. 15, from 10:30 to 12:30 at the usual place.
  • It is cumulative, although emphasis is put in the material covered after the second test.
  • You can use books, notes, etc as before.
  • Bring your own pens, pencils, calculators, AND PAPER. I won’t have extra paper if you don’t bring enough, and the margins of the exam won’t suffice. Mark with your name every single page you turn in.

See you Monday. Good luck!

Update [Dec. 22/08]:

  • Here is the exam for Calculus II – 175, and here are the solutions. (Silly typo in the solution of problem 4 corrected.)
  • Here is the exam for Calculus III – 275, and here are the solutions
  • I will be in my office on December 19 from about 11 until about 1, in case you want to stop by and pick up your test.
  • I won’t be on campus until the Spring term. I’ll post my new office hours soon, in case you want to stop by and pick up your test once I’m back. I’ll keep the exams and homework sets I still have through the Spring term, and you can collect them at any time during office hours. Whatever remains once the term is over, I will then discard. 

175, 275 -Homework 12

December 1, 2008

Homework 12 is due Tuesday, December 9, at the beginning of lecture. The usual considerations apply. This is the last homework set of the term. Each exercise is worth 1 point. 

  • 275: Turn in the problems you still have pending. Notice that the homework does not cover the last few sections of Chapter 14. However, these sections are included in the final, so make sure you try (on your own) a few exercises as practice. 
  • 175: Do not use the solutions manual for any of these problems. Turn in the problems you still have pending. Also: Section 8.8, exercise 28; section 8.9, exercise 40; section 8.10, exercise 19. There is no homework on the additional topics we will cover, but they will be included in the final.

175, 275 -Homework 11 and suggestions for the week after Thanksgiving

November 18, 2008

Homework 11 is due Tuesday, December 2, at the beginning of lecture. The usual considerations apply.

In 175 we will try to cover this week until section 8.10, but probably won’t get that far. As mentioned last week,  we will also cover additional topics that the book doesn’t mention or doesn’t treat in sufficient detail, once we are done with 8.10. These topics are uniform vs. pointwise convergence (including Wierstrass test), the behavior of the p-series \displaystyle \sum_{n=1}^\infty\frac1{n^p} for p=2,3,\dots, and infinite products. I will distribute notes of the topics not covered in the book. If you want to read ahead, and would like some of the notes ahead of time, please let me know.

In 275 we will cover Chapter 14, probably this week we will cover until section 14.4. Particularly important are the notion of conservative field and Green’s theorem. Then we will continue with surface integrals and orientations, Stokes’s and the divergence theorems. This will probably take another week, maybe a bit more. If there is any time left afterwards, we will see Lagrange multipliers 

Homework 11:

175: Do not use the solutions manual for any of these problems.

  • Turn in the problems listed for Homework 10  that you still have pending.
  • Section 8.6. Exercises 8, 25, 28, 37, 60.
  • Section 8.7. Exercises 2, 4, 39-48. 

Besides the exercises you have pending from last week, there are 17 new problems. Turn in the exercises you have pending, and at least 10 of the new problems. The others (at most 7) will be due December 9, together with the additional exercises for that week. 

275:  

  • Turn in the problems listed for Homework 10  that you still have pending.
  • Section 14.1. Exercises 1-8, 12, 16, 29.
  • Section 14.2. Exercises 6, 8, 34, 41.
  • Section 14.3. Exercises 2, 6, 12, 17, 20, 34, 38.
  • Section 14.4. Exercises 2, 8, 18, 31-35.

Exercises 14.1.1-8 count as a single exercise. Besides the problems pending from last week, there are 23 exercises. Turn in at least 10 of these. The remaining problems (at most 13) will be due together with a few additional exercises on December 9.


275 -Positive polynomials

November 11, 2008

When studying local extreme points of functions of several (real) variables, a typical textbook exercise is to consider the polynomial

P(x,y)=x^2+3xy+3y^2-6x+3y-6.

Here we have P_x=2x+3y-6 and P_y=3x+6y+3, so the only critical point of P is (15,-8). Since P_{xx}=2 and the Hessian of P is 2\times 6-3^2=3>0, it follows that (15,-8) is a local minimum of P and, since it is the only critical point, it is in fact an absolute minimum with P(15,-8)=-63.

P being a polynomial, it is reasonable to expect that there is an algebraic explanation as for why -63 is its minimum, and why it lies at (15,-8). After all, this is what happens in one variable: If p(x)=ax^2+bx+c and a\ne0, then

\displaystyle p(x)=a\left(x+\frac b{2a}\right)^2+\frac{4ac-b^2}{4a},

and obviously p has a minimum at x=-b/2a, and this minimum is (4ac-b^2)/4a.

The polynomial P of the example above can be analyzed this way as well. A bit of algebra shows that we can write

\displaystyle P(x,y)=\left(x-3+\frac32 y\right)^2+3\left(\frac y2+4\right)^2-63,

and it follows immediately that P(x,y) has a minimum value of -63, achieved precisely when both x-3+3y/2=0 and 4+y/2=0, i.e, at (15,-8).

(One can go further, and explain how to go in a systematic way about the `bit of algebra’ that led to the representation of P as above, but let’s leave that for now.)

What we did with P is not a mere coincidence.  Hilbert’s 17th of the 23 problems of his famous address to the Second International Congress of Mathematicians in Paris, 1900, asks whether every polynomial P(x_1,\dots,x_n) with real coefficients which is non-negative for all  (real) choices of x_1,\dots,x_n is actually a sum of squares of rational functions. (A rational function is a quotient of polynomials.) A nonnegative polynomial is usually called positive definite, but I won’t use this notation here.

If Hilbert’s problem had an affirmative solution, this would provide a clear explanation as for why P is non-negative.

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175, 275 -Homework 10 and suggestions for next week

November 10, 2008

Homework 10 is due Tuesday, November 18, at the beginning of lecture. The usual considerations apply.

In 175 we will try to cover this week until section 8.7 at least; it is possible this won’t happen until next week. Sections 8.3, 8.4, and 8.5 all introduce important tests for convergence of series; make sure you understand the arguments being presented (rather than just trying to memorize the tests). The material in section 8.6 is particularly important (conditional and absolute convergence). We will also cover some additional material on the p-series \displaystyle \sum_{n=1}^\infty\frac1{n^p}

Next week we will continue from section 8.7 (or at whatever point in the chapter we find ourselves at that point) on. We will also cover a few additional topics that the book doesn’t mention or doesn’t treat in sufficient detail, once we are done with 8.10. If you want to read ahead, the topics we will cover are uniform vs. pointwise convergence (including Wierstrass test) and infinite products. I will distribute notes of the topics not covered in the book.

In 275 we will cover from section 13.6 on, but the emphasis will be on section 13.8, and the notion of Jacobian. We will also present a few notions from linear algebra to make sense of the general version of the chain rule. Afterwards, we will continue with Chapter 14, which contains the main results from this course you are likely to use in the future. Chapter 14 will take some time to cover. 

Homework 10:

175: Do not use the solutions manual for any of these problems. 

  • Turn in the problems listed for Homework 9  that you still have pending.
  • Section 8.4. Exercises 3, 12, 23, 26, 35, 38, 40.
  • Section 8.5. Exercises 4, 8, 10, 25, 31, 34, 44, 47.

Besides the exercises you have pending from last week, there are 15 new problems. Turn in the exercises you have pending, and at least 7 of the new problems. The others (at most 8) will be due December 2, together with the additional exercises for that week. 
275:  

  • Section 13.5. Exercises 4, 8, 19, 22, 30, 45.
  • Section 13.6. Exercises 6, 11, 24, 30.
  • Section 13.7. Exercises 13, 21, 37, 60, 78, 79.
  • Section 13.8. Exercise 1, 6, 16, 21.

There are 20 exercises this week. Turn in at least 10. The remaining problems (at most 10) will be due together with a few additional exercises on December 2.


275 -Average values of harmonic functions

November 6, 2008

I want to mention here an important property of harmonic functions, the mean value property, and some of its consequences. I restrict myself to functions of two variables for clarity. 

Many important properties of harmonic functions (and, by extension, of analytic functions) can be established solely in the basis of the mean value property. I don’t know how to prove this property (or that it characterizes harmonicity) without appealing to Stokes’s theorem, or one of its immediate consequences (the topic of Chapter 14 of the book); in fact, I doubt such an approach is possible. It is a good exercise to see, at least formally, how this result gives the mean value property, but a rigorous treatment tends to be somewhat involved. Unfortunately, the arguments that show that the statements below hold tend to require techniques that are beyond the scope of Calculus III, so I will skip them.

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175, 275 -Homework 9 and suggestions for next week

November 3, 2008

Homework 9 is due Tuesday, November 11, at the beginning of lecture. The usual considerations apply.

In 175 we will try to cover this week until section 8.4 at least, but probably we won’t get there until next week. The key section here is 8.2; make sure you understand the notions discussed in 8.2 before going further. If you want to read ahead from 8.4, continue with sections 8.5 and 8.6; the difference between conditional and absolute convergence is very important here.

In 275 we will cover from section 13.4 on, and the goal is to reach 13.8, which probably won’t happen until next week or even the one after if things do not go well. Besides these topics, I will discuss the `mean value property’ of harmonic functions.

Homework 9:

175: Do not use the solutions manual for any of these problems. 

  • Section 8.1. Exercises 86, 88, 127. Also, the following exercise:

Starting with a given x_0, define the subsequent terms of a sequence by setting x_{n+1}=x_n+\sin(x_n). Determine whether the sequence \{x_n\} converges, and if it does, find its limit. More precisely: You must indicate for which values of x_0 the sequence diverges, and for which it converges, and for those that converges, you must identify the limit, which may again depend on x_0. You may want to try studying the sequence with different initial values of x_0 (choose a large range of possible values) to get a feeling for what is going on. 

  • Section 8.2. Exercises 14, 22, 38, 40 (do not use a calculator for this one; you can use that 2<e<3<\pi if necessary), 64-68, 71.
  • Section 8.3. Exercises 26, 35, 41, 43, 44.

There are 19 problems in total. Turn in at least 10. The others (at most 9) will be due November 18 together with a few additional exercises for that week. I suggest you start working on these problems early, as some may be a bit longer than usual.

275:  

  • Section 13.2. Exercises 5, 17, 39, 47, 54.
  • Section 13.3. Exercises 13, 16, 21.
  • Section 13.4. Exercise 27, 36.

175, 275 -Midterm 2

November 2, 2008

The second exam was Friday, October 31, during lecture. This exam is worth 10% of the total grade, and replaces the grade of the first exam if the new score is higher. As before, books, notes, and calculators were allowed. 

Failure to take the exam was graded as a score of 0. 

175: The midterm covered Chapter 7 of the textbook (and assumed knowledge of Calc I).

275: The midterm covered Chapter 12 of the textbook (and assumed knowledge of Calc I and II).

Below is a graph of the function in problem 5 of the 275 midterm.

z=x cos(y)

 

 


275- Constant curvature

October 23, 2008

The text we are using for Calculus III introduces the notion of unit tangent vector, principal unit normal vector, and curvature, for smooth curves \vec r(t), and it also mentions that circles and lines are planar curves of constant curvature. (A curve is planar if its image is contained in a plane, i.e., if it describes a two-dimensional trajectory.)

Surprisingly, though, the text does not explain (not even in the exercises) that circles and lines are the only smooth planar curves of constant curvature. The argument for this is simple enough, so I will show it here:

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