| Sections | Pages | Problems to do | Submit | Due date | Comments |
|---|---|---|---|---|---|
| 1-2 | 4-5 | 1-4,6,7,9,10 | None | None | |
| 4 | 11 | 1,4 | None | None | |
| 5 | 13-14 | 1,2,3,11,15,16 | 5,8 | Tuesday, January 26 | |
| 6-7 | 21-22 | 1,2,4,5,6 | 10 | Friday, January 29 | |
| 8-9 | 28-29 | 1,2,6,7 | 4 | Friday, January 29 | |
| 11-13 | 42-43 | 1,3,4,6,7,8 | None | None | |
| 28 | 89-90 | 1,6,7,10 | None | None | |
| 33 | 103-105 | 6,9,10,15 | None | None | |
| 14-17 | 53-54 | 1,2,3,4,5,6(a),7,10,11 | 9 | Monday, February 15 | For Problem 9, you might find it helpful to first rephrase the statement to be proven in a standard "if-then" form. |
| 10 | 31-32 | 2,4 | None | None | |
| 18-19 | 59-60 | 1,3,4,5,6,8 | 7 | Friday, February 19 | |
| 20-22 | 68-70 | 1,2,3,5,10 | None | None | |
| 20-22 | 68-70 | 4,7,8 | 9 | Thursday, February 25 | |
| 23-24 | 73-74 | 1,2,3,4,5,7 | 6 | Friday, February 26 | |
| 25 | 78-80 | 1,3,4,7,8 | None | None | |
| 94-95 | 350-351 | 1,2,3,4 | None | None | |
| 28 | 89-90 | 3,4,8,12 | None | None | |
| 33 | 103-105 | 8,13 | None | None | |
| 29-30 | 94-95 | 1,2,4,5,6,7,9 | None | None | |
| 31 | 96-97 | 1,3,4 | None | None | |
| 32 | 99-100 | 1,2,3,5 | None | None | |
| 36-37 | 115-116 | 1,2(a,b),3,4,5 | None | None | |
| 38 | 120-122 | 2,5 | None | None | |
| 39-40 | 128-130 | 1,2,3,4,5,6,7,10 | 11 | Friday, March 12 | |
| 41 | 133-134 | 1,3,4,5 | None | None | Problems 4 and 5 are examples of a way we will use the ML bound later in the course. |
| 42-43 | 141-142 | 2,4,5 | None | None | |
| 44-46 | 153-156 | 1,2,4,6 | None | None | |
| 47-48 | 162-164 | 1(a,b,c,d,e),2,3,4,6,7 | None | None | |
| 49-50 | 171-173 | 1 | None | None | |
| 51-52 | 181-182 | 1,2,3,4,6 | None | None | |
| 53-54 | 188-190 | 2,3,5,7,10,11 | 13 | Thursday, April 15 | |
| 55-56 | 198-200 | 1,2,4,6 | 7 | Friday, April 16 | |
| 62-63 | 230 | 1,2 | None | None | |
| 65 | 233-234 | 1,2 | None | None | |
| 66-67 | 238-239 | 1,2,3,5 | 4 | Monday, April 26 | |
| 68-69 | 245-247 | 2,3,4,5,8 | 7 | Tuesday, April 27 | |
| 64 | 230 | 3 | None | None | |
| 66-67 | 238-239 | 6 | None | None | |
| 71-72 | 257-259 | 3,5,9 | 4 | Tuesday, May 4 | |
| 73-74 | 265-267 | 1,5,7 | None | None | Last problem set! |
If you notice any broken links or if the daily note and assigned problems have not been updated in a timely fashion, send me a note so I can make corrections or updates as needed.
I will be available on a drop-in basis during office hours listed below. I'm also happy to schedule appointments for other times I am free. If you'd like to arrange an appointment, send me an e-mail or give me call (×3567).
Availability for Thursday, May 6:
Availability for Friday, May 7:
Availability for Monday, May 10:
Availability for Tuesday, May 11:
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Availability for Thursday, May 13:
Exam #5 is take-home. It is due on by 2 pm on Friday, May 14 (the end of the scheduled final exam period for the course). Let me know if you have questions of clarification or spot any potential typos on the exam.
Today, we looked at one last idea for using complex analysis techniques to evaluate real-valued improper integrals. I've assigned a few problems from Section 73 that involve this idea. (Note: For these assigned problem, you do not need the idea discussed in Section 74.)
In class, we finished one example and then did a second example of using complex analysis techniques to evaluate improper (real-valued) integrals. Looking back on the second example reveals that a bit of creativity is useful in selecting an appropriate contour integral to play with.
Today, we started in on an example of using complex analysis techniques to evaluate an improper (real-valued) integral. So, we start with a given improper integral and then find a related contour integral, usually involving at least part of a circle of radius R. We then look at the contour integral in two different ways:
By comparing the results of these two approaches in the limit as R goes to infinity, we can extract a value for the improper integral.
In class, we looked at a method of evaluating contour integrals that can be thought of as involving the "residue at infinity". The text's approach to this idea differs from the one we took in class but the result is the same. I've assigned problems from Section 64 and Sections 66-67 that involve this technique. At the end of class, I was a bit careless with signs in accounting for residues from points "outside" the contour. I'll write up a brief note on how to more carefully handle the details. For the assigned problems, this will not be an issue since these involve only the contribution from the "residue at infinity".
We spent the class session working out the details of Problem 5 from Sections 68-69. Pushing through all of the details leads to a nice way of proving a cool result. For reference, here's an outline of the big picture for the problem:
In class, we looked at problems from Sections 66-67 and then did an example using the techniques from Sections 68-69. I've assigned problems from Sections 68-69 that you should look at before Monday's class.
Since you have been focussed on the take-home exam, you might not have spent much time with recently assigned problems. To give you a jump-start on these, we worked on selected problems in small groups.
In class, we looked at functions of the form f(z)=p(z)/q(z) and we related the "poleness" of f to the "zeroness" of p and q. We used this relationship to find a nice formula for computing a residue for the special case in which p is nonzero at z0 and q has a zero of order 1 at z0.
I have assigned problems from Sections 62-63, 65, 66-67. Knowing that you might be busy with other things for the course right now, I'll plan to spend some time in class on Thursday making sure we know how to at least get started on the problems.
Today we looked at classifying an isolated singular point as a removeable singularity, a poles of order m, or an essential singularity. For a pole of order m, we have a theorem that allows us to separate out the singular factor in the denominator and push everything else into a nice function in the numerator.
We have skipped over the idea in Section 64 for now. We'll come back to this soon.
Exam #4 is take-home. It is due on Thursday, April 22.
Today, we had our first look at the residue theorem. This result provides us with a powerful way to evaluate a certain kind of contour integral. The key points in proving the residue theorem are (1) recognizing a Laurent coefficient b1 as a contour integral of f itself (for a suitable contour) and (2) an argument very similar to that used in proving the contour deformation theorem.
In the example we did in class today, we extracted the needed residues by finding the relevant Laurent series. Next week, we'll develop efficient methods for computing residues directly without finding the full Laurent series.
In class, we looked at the analogy Taylor series are to power series as Laurent series are to "extended power series". The Taylor series theorem tells us that if a function f is analytic on an open disk, then the Taylor series (based at the center point of the disk) for that function converges to f on the disk. The Laurent series theorem tells us that if a function f is analytic on an open annular region, then the Laurent series (based at the center point of the annular region) for that function converges to f on the annular region. We can view the Taylor series theorem as a special case of the Laurent series theorem in the following way: If f is analytic on a disk, then the Laurent coefficients bk will all equal zero by Cauchy's theorem and the Laurent coefficients ak will reduce to the usual Taylor series coefficients by the Cauchy Integral Formulas.
We will skip over the material in Sections 57-61 and move directly to the ideas in Chapter 6. In class, I'll give you a brief sense of the main ideas in the sections we'll be skipping.
Exam #4 will be take-home. It will be distributed tomorrow (afternoon most likely) and due on Thursday, April 22.
Today, we looked at "extended power series" which include negative powers of z or z-z0. There are two separate convergence issues for an extended power series: one for the series of positive powers and one for the series of negative powers. Intuitively, convergence of the positive power series might require not getting too far from the base point z0 while convergence of the negative power series might require not getting too close to the base point z0. Thus, we might expect the region of convergence to be an annular region centered at the base point.
A Laurent series is to an extended power series as a Taylor series is to a power series. That is, a Laurent series is an extended power series with coefficients defined in relation to a given function f (specifically by the integrals in (2) and (3) in page 191 of the text). The Laurent series theorem in Section 55 is analogous to the Taylor series theorem in Section 53. This theorem gives us conditions under which the Laurent series for a given function f converges to the function f.
A uniqueness result for extended power series allows us to find a Laurent series expansion for a given function without explicitly computing the Laurent series coefficients. If we can relate a given function f to a known series expansion (such as a geometric series), we can get the Laurent series expansion indirectly. We'll soon leverage this idea as an indirect means for evaluating contour integrals.
In class, we worked through a proof of the Taylor series theorem. This proof uses the Cauchy Integral Formula (CIF-0), the geometric series result, and the ML-bound.
Tomorrow, we will work out the nature and timing of Exam #4. We'll then address any question from Section 53-54 problems. Since I forgot to invite these questions today, I've changed the due date on the problem to be submitted.
In class, we looked at power series including the particular case of Taylor series. We stated and used the Taylor series theorem. On Monday, we'll work through a proof of the theorem after we address questions from assigned problems.
Today, we looked at series in the complex setting, including the important example of geometric series. As part of your homework, you should find two different ways to express 1/(3+z) as a geometric series. For each, you should give the domain of convergence (i.e, the set of z values for which equality holds). In addition, I've added two problems to the original assignment for Sections 51-52.
In class, we look a sequences in the complex setting and then began looking at series in the complex setting. We'll continue with series on Thursday as we work our way toward Laurent series.
I've assigned a few problems from Sections 51-52 that deal with sequences. I'll assign additional problems from these sections that deal with series after Thursday's class.
Today, we used Liouville's theorem in a complex-analysis proof of the fundamental theorem of algebra. Purely algebraic proofs of this theorem are not easy.
We will skip over Section 50 for now, leaving it as an option to come back to at the end of the semester as time permits. So, our next goal is to master sequences, series, and power series in the complex setting. Our main goal here is to build up to a generalization of power series called Laurent series.
Today, we looked at several results that shows important ways in which complex-valued functions differ from real-valued functions. Perhaps the most striking is Liouville's theorem: If f is entire and |f| is bounded for all z, then f is constant. A simple example such as f(x)=cosx shows that the analogous result is not true for real-valued functions. On Monday, we'll put Liouville's theorem to work in a complex-analysis proof of the fundamental theorem of algebra.
In class, we discussed the generalized Cauchy integral formula (which we denoted CIF-n). We started with an informal (i.e., reasonable but not justified) way to guess CIF-1, then gave an argument with some more rigor, and finally outlined a fully rigorous proof. We then used the informal method to conjecture CIF-n (given as Equation (4) on page 161 of the text).
Exam #3 will be on Thursday, April 1 and will include a small take-home piece that will be due on Monday, April 5. For the in-class piece, you can use the 80 minute period from 11:00 to 12:20 that the classroom is available.
Exam #3 will focus on material we've covered since the previous exam. From the text, this is material in Sections 36 through 48 with the exception of Section 45. (Section 45 has a direct proof of the Cauchy-Goursat theorem that is not based on Green's theorem.)
In class, we went through an informal argument for and then a careful proof of the Cauchy integral formula. One immediate use of the Cauchy integral formula is in evaluating contour integrals of a certain form. We did one example of this in class and you'll get practice with other examples in the assigned problems from Section 47-48.
On Monday, we'll generalize the basic Cauchy integral formula. As part of this, we'll be proving that if a function is analytic at a point, then all derivatives of the function exist and are analytic at the point.
Exam #3 will be on Thursday, April 1 and will include a small take-home piece that will be due on Monday, April 5. For the in-class piece, you can use the 80 minute period from 11:00 to 12:20 that the classroom is available.
We need to schedule Exam #3 for Thursday or Friday of next week. On Thursday, we could take advantage of the 80 minute period the classroom is available (from 11:00 to 12:20). We'll discuss the options and make a decision in class tomorrow.
Today, we arrived at a version of Cauchy's theorem by using Green's theorem for line integrals. In the argument we developed, we did not focus on some technical conditions needed in order to use Green's theorem. In particular, using Green's theorem in the way we did requires that the derivative f′ be continuous. Section 45 has a proof of Cauchy's theorem that does not require this continuity assumption on f′. We will skip over this proof for now and focus on more consequences of Cauchy's theorem. If time permits later, we'll step through an outline of the proof in Section 45.
We are reviewing Green's Theorem in order to provide a relatively easy proof of Cauchy's Theorem. We'll get to this tomorrow.
Today, we played with iterating complex-valued functions. This is often referred to as a discrete dynamical system. Some dynamical systems have chaotic behavior. For example, iterating the squaring function f(z)=z2 is chaotic for points on the unit circle.
Our main goal today was to get some sense of how fractals make an appearance in studying iterations of a complex-valued function. Here, we looked at the family of functions fc(z)=z2+c where c is a complex constant. The Mandelbrot set is defined in reference to this family of functions. The Mandelbrot set lives in the c-plane and turns out to have a fascinating structure. There are lots of web resources for exploring that structure. Here are a few that I have found useful and interesting:
We looked at the first two of these in class today. Each of the others does similar things with different feature sets. (The Boston University site has not been working when I've checked it over the last few days.)
All of this is for your own cultural benefit. You will not be accountable for these ideas as part of the course.
Enjoy your break!
Today, we stated the ML bound for contour integrals and then used it in finishing a proof of the antiderivative/path independence theorem.
We are now set up to take on the most theorem of the course, namely Cauchy's Theorem. (Our text refers to it as the Cauchy-Goursat Theorem. The more common name is Cauchy's Theorem.) We'll get to this after Spring Break. Tomorrow, we'll explore the way in which fractals and chaos show up through iterating complex-valued functions.
In class, we stated the main theorem from Section 42 and started in on the proof. To finish off the most difficult leg of the proof, we need a useful result that we will call the ML bound for a contour integral. We'll look at this in class on Thursday before finishing off a proof of the theorem on antiderivatives and path independence.
Here's the link of the day.
In class, we went a bit more carefully through some of the technicals details needed to define contour integral. In particular, we looked at calculus for "hybrid" functions w:R→ C and we defined what we mean by contour. Tomorrow, we'll finish off the discussion we started at the end of class about reparametrizations.
Today, we introduced contour integrals for complex-valued functions. Most of the topics we discuss for the remainder of the semester will connect to contour integrals in one way or another.
After a bit of motivation, we jumped right into a definition of contour integral without pausing to consider a few technical issues. Our definition involves parametrizing a curve C with a function γ(t) defined for t=a to t=b. The contour integral of f(z) over C is then defined as the integral from a to b of the function f(γ(t))γ′(t). In jumping to this definition, we did not stop to consider
The text considers these issues in Sections 36-37 (on working with w:R→ C) and Section 38 (on defining the class of curves that we will refer to as contours).
The basic idea in Sections 36-37 is straightforward: We define the integral of w(t) from t=a to t=b as the integral of Re[w(t)] plus i times the integral of Im[w(t)]. This allows us to bring in all of the familiar properties of integrals for real-valued functions. I've assigned a few problems from these sections.
On Monday, we'll briefly discuss the highlights of Sections 36-37 and Section 38.
Exam #2 will be on Thursday. It will cover material from Sections 10, 14-25, 28-33, and 94-95. Note that we have skipped the material in Sections 26 and 27 for now.
In class, we finished discussing complex logarithms. If we account for different branches, most familiar properties of logaritms generalize in a natural way. We then defined complex exponents in terms of the complex exponential and logarithm maps. I have assigned problems from Sections 29-32. Many of these are computational rather than proof-construction.
Exam #2 will be on Thursday. It will cover material from Sections 10, 14-25, 28-33, and 94-95. Note that we have skipped the material in Sections 26 and 27 for now.
Toward the end of class, we begin discussing how to invert the exponential function. Since the exponential function defined on the entire complex plane is not one-to-one, we must first choose a suitably restricted domain. Any horizontal strip of height 2π will do. So, the inverse function we get depends on which strip we choose. For each choice, we get a branch of the complex logarithm map. Each branch is defined on a domain consisting of all points except those on a specific ray based at the origin. We'll continue discussing the logarithm map on Monday.
Exam #2 will be Thursday, March 4.
In class, we showed that an analytic function f maps a vector at a point z to a corresponding vector at the image point f(z) with a scaling by the factor |f′(z)| and a rotation by the amount arg(f′(z)). All vectors at the point z are scaled by the same factor and rotated by the same amount. Turning this around, we can say that if not all vectors are scaled by the same factor or rotated by the same amount, then the function f is not analytic at the point z. We used this argument to geometrically argue that the conjugation function is nowhere analytic.
Exam #2 will be Thursday, March 4.
In discussing Problem 5 from Sections 23-24, I lost the ability to articulate anything coherently. Here's an attempt to make up for this in writing. We are concerned with the function f(z)=2z-2+i=2(z+(-1+i/2)). In particular, we want to determine the points in the z-plane that are mapped to the negative real axis by f. In the factored form, we break the mapping effect of f into two steps: translation by -1+i/2 followed with scaling by a factor of 2. Consider the points on the ray z=x-i/2 with x≤1. Under the translation, this ray will be moved left 1 and up 1/2 to land on the negative real axis. The scaling will map the negative real axis to itself (with a uniform stretching). So, the net effect is that the ray z=x-i/2 with x≤1 is mapped to the negative real axis (including the origin).
In class, we defined the idea of analytic function. This concept will be central to much of what we do in the remainder of the course. We'll see many theorems that start with the hypothesis "If f is analytic..."
Today, we translated the C-R equations into polar form from cartesian form. The key tool for this translation is the chain rule. We worked out the details of one direction in a statement of equivalence. Problem 7 has you work out the details for the other direction of the equivalence. Note that we did the first part of Problem 7 near the end of class.
At the end of class, we wrote down a formula for computing the derivative of f=u+iv in terms of partial derivatives of u and v with respect to r. Problem 8 asks you to derive this formula starting from the cartesian version. Problem 9 asks you to come up with a formula for the derivative of f=u+iv in terms of partial derivatives of u and v with respect to θ.
In class, we worked through a proof of the theorem that gives sufficient conditions on the relevant partials derivatives to guarantee differentiability of a complex-valued function. This gives us a nice tool for analyzing differentiability without directly analyzing the limit of a difference quotient.
I've assigned problems from the Section 20-22 problems with focus on the ones relevant to the ideas in Sections 20 and 21. I'll assign problems relevant to the ideas in Section 22 after we've discussed those ideas in class tomorrow. I'll then also decide on any problems to submit.
We continued exploring differentiability and derivatives for complex-valued functions. In particular, we came up with the Cauchy-Riemann equations as a necessary condition for complex-differentiability. On Thursday, we'll turn things around to get sufficient conditions.
Today we looked at extended the ideas of differentiability and derivative from real-valued functions to complex-valued functions. The basic idea is the same: define these notions in terms of the limit of a difference quotient. As we'll see, this means that many of the familiar results and rules for differentiation generalize in a natural way from the real-valued world to the complex-valued world. There are, however, important ways in which complex-valued functions differ from real-valued functions.
In class, we developed some of the technical language from Section 10 of the text. We circle back to this section to pick up other pieces of terminology as we need.
We also introduced the idea of the extended complex plane which adds one point, labeled ∞, to the "regular" complex plane. With this, we can refer to ∞ as the value of certain limits and we can define limits at ∞.
Note that I have added two problems (10 and 11) to the list for Sections 14-17 and I have assigned some problems from Section 10.
In class, we looked at two examples of "epsilon-delta" proofs at a level that you should become comfortable reading. This level is a bit higher than what I will expect you to be able to construct on your own.
For convenience, here's the link for the scan of part of Chapter 2 of the text. Sections 14-17 are part of this scan.
We are working with a precise definition of limit (a.k.a. an "epsilon-delta" definition). Your minimal goal should be basic mastery of this definition by which I mean being able to read "epsilon-delta" proofs and being able to construct simple proofs on your own. Full mastery of "epsilon-delta" proofs is a central goal of the advanced calculus course (Math 321).
In class, we developed a definition for continuity at a point for a complex-value function. This definition of continuity is a special case of the more general idea of limit. We'll look at limits tomorrow. I'll then assign some problems from Sections 14-17. For now, you should think about how to draw pictures and give an algebraic proof for the fact that the function f(z)=conjugate of z is continuous for all points.
Exam #1 will be Friday, February 5. It will cover material from Sections 1-9 and 11-13 along with parts of 28 and 33. It will not cover material from Sections 28 and 33 that deals with calculus concepts.
Here's a scans of Section 28 and Section 33 from the text.
The text briefly introduces the complex exponential function in Section 13. Further details on the complex exponential function and on other elementary functions are put off until Chapter 3. In class, we're looking at some small pieces of Chapter 3 right now. In particular, we are pulling some ideas from Section 28 (on the complex exponential function) and from Section 33 (on complex trigonometric functions). I'll have these sections scanned and posted later this afternoon.
I have assigned a few new problems (#4,6,7) from Sections 11-13.
Exam #1 will be Friday, February 5.
The video Mobius Tranformations Revealed beautifully illustrates some specific examples of complex functions as mappings. In class, we explored the specific example w=1/z. If you are interested in reading more about the general class of Mobius transformations, check out Sections 86-88 of our text. (Note that our text refers to these as linear fractional transformations.) We might have time to return to these in more detail at the end of the semester.
Exam #1 will be Friday, February 5.
We've skipped over Section 10 for now. We'll come back to this soon. The material we are now discussing is from Sections 11-12 in the text.
After narrowly averting death (or, at the very least, serious injury), we developed some basic language and notation for complex-valued functions and then looked in-depth at one example of visualizing a complex-valued function as a mapping from the input plane to the output plane.
I've assigned some problems from Sections 11-13. I'll assign additional problems from these sections after the next class.
Here's a scan of part of Chapter 2 of the text. I'll remove this link in a few days once the textbook situation clears up.
Exam #1 will be Friday, February 5.
In class, we finished up some examples on computing roots of complex numbers. I've assigned problems from Section 8-9 related to this. Before class on Thursday, you should put reasonable effort into the assigned problems from Sections 6-7 and 8-9.
Our next task is to begin building some intuition for complex-valued functions. As a prelude to this, we looked at real-valued functions from the viewpoint of mappings (as opposed to graphs). On Thursday, we'll begin to see how this mapping viewpoint generalizes to complex-valued functions.
We started class addressing questions from Section 4 and 5 problems. One of these involved proving the triangle inequality. To be complete, we also proved what we will call the reverse triangle inequality and did an example using these to get upper and lower bounds on a certain distance. This is the stuff I forgot to take care of last Friday.
In the last few minutes of class, we looked at an example of finding roots of a complex number. We'll follow up on this tomorrow.
I've assigned probems from Section 6-7. We'll address questions from this in class tomorrow (and Thursday if needed).
I intended to start the "new material" part of class today with a few more ideas on inequalities. Somehow, that didn't happen. We'll take care of that on Monday, possibly as part of addressing questions from the Sections 4 and 5 problems.
One major theme is Sections 1-3 is that the algebraic rules for manipulating variables representing complex numbers are natural generalizations of the rules for manipulating variables representing real numbersi. All of this requires proof, much of which is straightforward and a bit tedious. In this course, we will not dwell on proving all of these algebraic facts. Reading the text and working on the assigned problems should give you a solid sense of how these proofs are constructed.
For Monday, you should bring questions on the assigned problems from Sections 4 and 5.
You should quickly become comfortable with the material in Sections 1-5 of the text. The individual ideas are generally straightforward.
We are developing multiple ways of looking at complex numbers, both algebraic and geometric. We can think of a complex number as a single object or we can think of it as composed of two parts (the real part and the imaginary part). In the long run, having multiple viewpoints available is an advantage. For example, one can often make a conjecture based on geometric thinking and then construct a proof algebraically. Being able to effectively put multiple viewpoints to use requires flexibility in thinking. You can develop your flexibility by forcing yourself to think in multiple ways. Don't stop if you find one successful approach to a problem; look for a second way to solve the problem. Then, step back and ask about the advantages and disadvantages of each approach. Does one generalize more readily? Is one more elegant or streamlined?
I've assigned problems from Sections 4 and 5. Note that two problems from Section 5 are to be submitted next Tuesday. We'll address questions from the Section 1-2 problems in class tomorrow and questions from the Sections 4 and 5 problems in class on Monday.
In class today, we looked at two approaches to the complex numbers:
We had a small taste of the first approach. We'll follow up with more detail on the second approach in class tomorrow.
I've assigned a few problems from Sections 1-2. You can start looking at these. Some will make more sense after Thursday's class. Later this week, I'll assign a problem or two to submit and give a due date.
Here's a scan of Chapter 1 of the text. I'll remove this link in a few days once the textbook situation clears up.
The Mathematical Atlas describes the many fields and subfields of mathematics. The site has numerous links to other interesting and useful sites about mathematics.
If you are interested in the history of mathematics, a good place to start is the History of Mathematics page at the University of St. Andrews, Scotland.
Check out the Astronomy Picture of the Day.