Fabian Waleffe -- Math 321 -- U. of Wisconsin

Math 321 homework, lecture pointers and learning objectives: Fall 2010

Fall 2010, Math 321 TA page

PART 1. Vector algebra: geometric concepts and index notation

HOMEWORK 1 to 10 on PART 1

PART 2. Vector Calculus: curves, surfaces, volumes, gradient, divergence, curl, ...

HOMEWORK 11 to 20 on PART 2

PART 3. Complex Calculus

Numbered exercises refer to the Complex Calculus notes on the Math 321 home page that you have been studying. There are no problems to hand in. You should work on the problems listed below in preparation for exams. `No pain, no gain'. If you get someone else to solve the problems for you, you are not learning. If you are not struggling with some of the material, you are not learning.

Homework 21, Tue. 11/30/10 : Functions of one complex variable Section 1.1-1.8: Complex numbers, geometric series, Taylor series, exponential, log, roots,...
1. Old basic stuff and complex jargon:
  - What is a complex number? how to add, multiply -- and divide complex numbers
  - What is the real part of i ? What is the imaginary part of i ?
  - What is a `complex conjugate'?
  - What is the `modulus' of a complex number?
  - What is the `argument' of a complex number?
  - What are all the z's such that |z-(1+2i)|=1? Sketch in the complex plane.
2. 1.1. Old basic algebra and series stuff:
  - What is a geometric sum? How do you derive the compact formula for the geometric sum?
  - What is a geometric series? For what complex numbers does it converge? diverge?
  - What happens when |q|=1? Try it out for a few different such q's
  - If you borrow A dollars at annual interest rate R, compounded monthly for a term of N years, what are your monthly payments? (1) What is the monthly interest rate? (2) If you borrow A at month 0 how much do you owe in N years compounding monthly? (3) If you put P into an account that pays you the same interest rate (quoted as an annual rate R but compounded monthly) every month starting at month 1, how much will you have after N years compounding monthly? Explain how to determine the monthly payments P, that is: derive the formula for P in terms of A, R and N . What does this have to do with this section?!!
  - Exercises: 1,2,3.
3. 1.3.
  - Prove formula (8) using geometric sums.
  - Exercises: 1,2.
4. 1.4. What is a useful convergence test for series? Why does that test work? Is it sufficient for convergence? Is it necessary for convergence? (explain)
5. 1.5.
  - What is a power series?
  - Where in the complex plane does a power series converge? where does it diverge?
  - What is a Taylor series? You know and understand formula (17), don't you?
  - What are the Taylor series of exp(z), cos(z) and sin(z)? You know those, don't you?
  - What is Euler's formula? do you know it?, you must! can you derive it?
  - Exercises: 1--7
6. 1.6. Really basic section, lots of "2+2" here. Do you know what 2+2=? is that important?
  - You know formula (22), (23), (24), (25), (26), (31), (32) and (33), don't you? If not, you really should! You saw all these for a real variable x, and they apply likewise for a complex variable z.
  - What is the polar form of a complex number? for what operations is the polar form particularly convenient?
  - Exercises: 1--6.
7. 1.7. OK, now it's really getting complex...
  - You understand that formula (39), (40) and (41) directly follow from Euler's formula (33) so you know them, don't you?
  - Digest the discussion between (39) and (40).
  - In Matlab, arg(z) is written "angle(z)". Fire up Matlab and compute "angle(1+i 0.0001)" and "angle(1-i 0.0001)" where i²=-1. Since those two numbers (1+i 0.0001) and (1-i 0.0001) are very close to each other, shouldn't their angles be close also?
  - In Matlab plot "angle(6 exp(i t))" where i²=-1 as a function of t for t=0 to 20. For steps dt=1/16 that's
    >> t=[0:1/16:20]; i=sqrt(-1); plot(t,angle(6*exp(i*t)))
    What is angle(6*exp(i*t) ≡ arg(6 exp(i t))=?
8. 1.8. Huh?! Can't we just eat turkey and pumpkin pie and watch football everyday...? Nope, chew on this now:
  - OK, you know e^z for any complex z, AND you know both the cartesian and the polar representations of complex numbers. Now find all the z's such that e^z=a where a is an arbitrary complex number. For instance find all the z's such that e^z=3+2i.
  - What is the log of a complex number? What is ln(i)? What is ln(-1)?
  - What is a complex exponential? What is iⁱ?
  - What are the `roots of unity'? i.e. Find z s.t. zⁿ=1 where n is an integer.(yep, many of you played a lot with this in Math 319 and/or math 320, didn't you?)
  - What is `factoring a polynomial'? Can you always do that?
  - Exercises: 1, 2, 3(i) and (ii). (if you got here without too much trouble, look at (iii) it's cool, otherwise forget about (iii), you've got other fires to attend to).
  - You know and understand formula (46), (47) and (50), don't you? That's "2+2" also. OK, it's more like 2² or (2+1i)⁽³⁺²ⁱ⁾ = ??.
Thur. 12/02/10 Jacob Schmid Interlude: What do we do with R, the radius of convergence?
Homework 22, Thur. 12/02/10 : Functions of one complex variable STUDY Section 2: complex differentiation, Cauchy-Riemann, orthogonal coordinates, conformal mapping.
1. What are the Cauchy-Riemann equations? where do they come from? What do they mean geometrically? What is the relation with Laplace's equation?
2. Is f(z)=|z| differentiable with respect to z? Explain/prove. Can you explain/prove without doing any calculations? (hint: what is a complex derivative? where is |z| constant?)
3. Is f(z)=conj(z)=z* differentiable with respect to z?
4. If we consider the mapping z → w=f(z) from the complex z=x+iy plane to the complex w=u+iv plane, when are angles preserved? What does it mean that angles are preserved? How do you show/prove that angles are preserved? Where are angles not preserved?
5. Exercises 2.2.1, 2, 3. You might want to look at the complex map examples, after having tried to figure out the problems for yourself, of course!
6. Intro to visualization of complex functions using Matlab Matlab commands ready for `cut-and-paste' + output.
Homework 23, Tue. 12/07/10 : Functions of one complex variable Section 3: complex integration
1. Must know and understand (i.e. know how to derive/justify): (1) Meaning of an integral in complex plane, (2) how to explictly compute an integral using a curve parametrization (e.g. 2 examples on page 15), (3) Cauchy's Theorem
EXAM 3, THURSDAY 12/09/10, 11:00-12:15pm ALL Material on complex variables, `elementary' complex functions (including exponential, sine, cosine, log, complex exponentials and roots), and functions of one complex variable, up to and including Cauchy's theorem for integration.
Homework 24, Tue. 12/14/10 : Complex integration, Sections 3 and 4.
1. Must know and understand (i.e. know how to derive/justify): (1) Meaning of an integral in complex plane, (2) how to explictly compute an integral using a curve parametrization (e.g. 2 examples on page 15), (3) Cauchy's Theorem: What is it? when does it apply? (4) Justification of `contour deformation' (picture and argument on page 16), (5) application to integral of (z-a)ⁿ, n=0,+/- 1, +/- 2, ... , (6) Cauchy's formula and its derivation (pg. 18), (7) Cauchy's general formula and at least a quick and dirty justification/derivation (eqns. (74), (79)) as well as rederiving (74) ≡ (79) by expanding the numerator f(z) in a suitable Taylor series, as done at the end of class.
2. Exercises: 3.1. 1, 2, 3, 4.
3. Section 4: Study `first, 2nd and 3rd key example (not) done in class' (but solved in the notes). We discussed (87) in class.
4. Exercises: 4. 1, 2, 3.
5. You can stop at eqn (88). You can also study beyond that, but you won't be tested on material beyond eqn (88). It's too bad. 4th `key example' and exercises 4.4 and 4.5 are classics. And the little branch cut problem on page 26 is nice. Maybe you can study this over the break...