Homework problems and due dates (not the dates the problems are assigned) are listed below. This list, especially the due dates, will be updated frequently, usually in the late afternoon or evening the day of class or the next morning. Due dates, and assignments more than one lecture ahead, are estimates; in particular, due dates may be moved either forward or back, and problems not currently on the list from a given section may be added later (but prior to their due dates, of course). Note that on a given day there may be problems due from more than one section of the book.Exam dates and some miscellaneous items may also appear below.
If one day's assignment seems lighter than average, it's a good idea to read ahead and start doing the next assignment, which may be longer than average.
Unless otherwise indicated, problems are from our textbook (O'Neill, Elementary Differential Geometry, revised 2nd edition). It is intentional that some of the problems assigned do not have answers in the back of the book or solutions in a manual. An important part of learning mathematics is learning how to figure out by yourself whether your answers are correct.
Read the corresponding section of the book before working the problems. The advice below from James Stewart's calculus textbooks is right on the money:
Some students start by trying their homework problems and read the text only if they get stuck on an exercise. I suggest that a far better plan is to read and understand a section of the text before attempting the exercises.
Date due Section # / problem #s W 1/11/12 Show that if real-valued functions f and g on R3 are infinitely differentiable, then so are f + g and fg.
1.1/ 1-4 F 1/13/12 Do all the homework I assigned in class. (This is always a part of your HW, implicitly, even when I don't say it explicitly on this webpage.) 1.2/ 1-5 W 1/18/12 1.3/ 1a, 2abc, 3-5 Read section 1.4 (although I will still do most of this in class on Wednesday), and get started on the problems due Friday so that you don't have so many to do between Wednesday and Friday.
F 1/20/12 1.4/ 1-5.
In #4, "log" means "natural log", i.e. loge (what you're used to calling "ln"). Except when teaching calculus, differential equations, and lower-level courses, most mathematicians use the notation "log" to mean loge, and use the notation "log10" for the base-10 logarithm in the rare instances that they want to refer to this function. Older calculus textbooks also use "log" to mean loge. Base-10 logarithms have essentially no use in higher mathematics, so the default base of "log" is the one of greatest use, e.M 1/23/12 1.4/ 6-9
Problems 6 and 7 should be thought of as a pair. The point of problem 7 is to drive home what you proved in problem 6. After doing 7a, use problem 6 to predict the answers to 7b. Then do 7b and check your predictions.
- Let a,b be real numbers, with a >0, and let α(t) = (a cos t, a sin t, bt), —∞ < t < ∞. In class we saw that the curve C parametrized by α is a helix if b ≠ 0, and a circle if b = 0. Show that α is a regular parametrization of C (regardless of the value of b).
- Let I be an open interval, β : I → R2 a smooth curve in the plane. Thus β(t)=(β1(t), β2(t)), where the βi: I → R are differentiable. Define α : I → R3 by α(t)=(t, β1(t), β2(t)). The Curve C in R3 parametrized by α is called the graph of β. Show that α is a regular parametrization of C, regardless of whether β is regular.
Note that if we identify R × R2 with R3, by identifying (x, (y,z)) with (x, y,z), then the definition of α can be written more simply as α(t)=(t, β(t)), and C is exactly the set of points {(t, β(t)) : t∈ I}. The similarity of this with the definition of "graph of a real-valued function of one variable," is the reason C is called the graph of β.W 1/25/12 Read Section 1.5. In Monday's class, we got about halfway through it, but I'd like you to get started on more of the HW problems than could be done based just on how far we got in class. I will still cover a bit more of this section in class on Wednesday, but probably not all of it. I'll still want you to know everything there, even if I don't spend class time on it. Most of what I'd omit from lecture is pretty mechanical. 1.5/ 1-7 F 1/27/12 (I meant to make this part of the previous assignment.) This is a continuation of the non-book problems 1 and 2 that were due Monday 1/23. Show that the Curve-parametrizations α of a helix (the case b ≠ 0, in problem 1) and a general graph (in problem 2) are proper. 1.5/ 8-10. In #10, "continuation" refers to what's being shown in problems 9 and 10 for general functions f; you're not continuing with the specific function f of problem 9. Also, for what you're asked to prove here, it's not sufficient to say, "This is true because of what we learned in Calculus 3." Chances are, proofs were not emphasized in your Calc 3 class, so you may have just been told "Believe this, it's true." If you were fortunate enough to have an instructor who did prove this fact about max/min, it still will not harm you to write down the proof again. Suggestion: proof by contradiction. I may ask you to turn in problems 8, 9 and/or 10.
1.6/ 1a, 5 (read Example 6.1, which continues onto p. 30, first), 6, 9. In the setup of problem 1.6/ 8 (which is part of the next assignment, not this one) you'll see a 1-1 correspondence "(1)" between vector fields and 1-forms, and a 1-1 correspondence "(2)" between vector fields and 2-forms. These correspondences are also valid at each point p of R3, if you replace vector fields, 1-forms, and 2-forms, respectively, with tangent vectors at p, cotangent vectors at p, and what I called "2-covectors at p" in class.
Let vp, wp be tangent vectors at p ∈ R3, let φp, ψp be the corresponding cotangent vectors using correspondence (1), and let up ∈ Tp R3 be the covector corresponding to φp ∧ ψp. In terms of the "classical" vector operations you learned in Calculus 3 (and/or physics classes), give a simple formula for u in terms of v and w. (Of course, if we let the point p vary, the analogous formula holds for vector fields.)M 1/30/12 Read any part of Section 1.6 that you haven't already read. Theorem 6.4(c) is what I was proving at the end of class on Friday. The calculation in the book is straightforward and easy to follow, so I'll just let you read it there instead of spending more time on it in class. You'll prove Proposition that I put on the board, "d composed with d =0", in exercises 3 and 7. 1.6/ 1b, 2-4, 7, 8 Homework to be handed in. On Wednesday, Feb. 1, hand in problems 1.5/ 8,9,10 (which were part of the assignment due Friday 1/27). Before you start writing up anything to hand in, please make sure you read the rules for hand-in homework. W 2/1/12 1.7/ 1-5, 7. Read the instructions at the start of the exercises to see what map F the first four problems refer to. Read this handout (Jacobians, Directional Derivatives, and the Chain Rule). For your class, the exercises are optional, but the more you do, the more you'll learn. As you'll see, these are notes that I wrote for a linear algebra class many years ago, but they should help you better understand some of what we did in Monday's class, and will be continuing in Wednesday's class. Some of the notation, of course, is different, and all the vectors in the handout are un-based. (Not new): Wednesday is the hand-in day for 1.5/ 8,9,10. F 2/3/12 Read anything in Section 1.7 you haven't read yet. In class, so far we've covered everything except what's on p. 40, plus some extra. (Some of the extra was problems 8 and 9b at the end of the section. In 9b, I'm not sure why O'Neill says that, except in low dimensions the formula "becomes formidable when expressed in terms of Jacobian matrices". It becomes formidable if you write out the entries of the product of the Jacobians of F and G. But in terms of the matrices themselves, the formula
JG o F (p) = [JG (F(p))] [JF (p) ]
is fairly simple.)Recall the following fact from linear algebra: if V and W are finite-dimensional vector spaces of the same dimension, and L1: V → W and L2: W → V are linear transformations, and L2 o L1 is the identity map V → V, then both L1 and L2 are isomorphisms, and each is the inverse of the other. (This fact is not as obvious as it may appear. Part of what this fact is saying is that, under the given hypotheses, L2 o L1 is the identity map of W. Without the assumptions that L1 and L2 are linear and that dim(V)=dim(W), it is not true that "L2 o L1 = identity map of V " implies "L1 o L2 = identity map of W.")
Now suppose that O1 and O2 are open sets in Rn, that F: O1 → O2 and G: O2 → O1 are differentiable maps (not assumed linear!) for which G o F is the identity map of O1, and that p ∈ O1. Using the fact recalled in the paragraph above, and the result of problem 1.7/5, prove that F*p : TpRn → TF(p)Rn is an isomorphism, and that G*F(p) = (F*p)–1.
(The fact you are proving in this problem, in addition to being important in its own right, should help you do problems 1.7/ 6b and 9c.)1.7/ 6, 9c, 10 M 2/6/12 Read Section 2.1, about half of which we covered in class on Friday. I may talk briefly about orthogonal matrices (p. 48) on Monday, since not all of you may be familiar with that. But most of the remainder is on cross-product, which we transfer to an operation on each tangent space TpR3 just as we did with dot-product. Since you've all seen in Calc III, I think that the review in Section 2.1 should be enough, and I think our class time would be better spent moving on to the next section. Section 2.2 will still be mostly review of material you may remember from Calc 3, but the review will be important. In Sections 2.3 and 2.4, some of the material there will be familiar to you, but the ratio of new material to review increases. From Section 2.5 on, almost everything will be new. 2.1/ 1-7,9-11 W 2/8/12 2.1/ 12 2.2/ 4 F 2/10/12 2.2/ 1,3,6,10,11 Exercise 2.2/ 5 is not correctly stated. Figure out what's wrong with it, and solve a correctly re-stated version of the problem. (a) Let f : R→R, and define α : R→R3 by α(t) = (f(t) cos(t), f(t) sin(t), f(t)).
If f is monotone (increasing or decreasing), the Curve parametrized by α can reasonably be called a "conical helix". Figure out why.
For the rest of this problem, f and α are as above.
(b) For such a curve-parametrization α, write down (in terms of f ) the integral that gives the arclength function s based at t=0.
(c) Show that α is regular provided there is no a∈R for which f(a) = f '(a) = 0.
(d) For the case f(t) = et, sketch the Curve parametrized by α.
(e) Again for the case f(t) = et, find an explicit formula for s(t), solve for t in terms of s, and write down the corresponding unit-speed reparametrization β of α.
(f) What is the domain of β in part (e)? You should find that it is an interval of the form (–a,∞), where a > 0. What is the value of a telling you geometrically?
M 2/13/12 2.2/ 2 2.3/ 4,8 W 2/15/12 In class we computed the curvature and torsion of helices (or circles, if b=0) with parametrizations of the form α(t) = (a cos t, a sin t, bt), where a >0. We found that the curvature and torsion are constant and are given by
κ = a/(a2 + b2) , τ = b/(a2 + b2).Solve the above equations for a and b in terms of κ and τ (you should find that there is a unique solution-pair (a,b) for each pair (κ,τ) with κ > 0), and substitute back into the equation for α(t) to see that this parametrization can be re-expressed purely in terms of the curvature and torsion. Thus, within the class of curves that have a parametrization of this form, the curvature and torsion completely determine the curve. In a few lectures, we will ask and answer a more general question: for general smooth curves with everywhere-positive curvature (not assumed constant), to what extent do curvature and torsion (also not assumed constant) determine the curve? You can ponder this in the meantime.
2.3/ 1-3,5-7. Note: In any problem in which the binormal B and/or the torsion τ appear, the assumption "wherever κ > 0 " is implicit.
Geometric interpretation of #5. For every v ∈ R3, the map Rv: R3 → R3 defined by Rv(w) = v ✗ w is linear. If v = 0 then Rv maps every vector w to 0, of course. If v ≠ 0, then every vector w can be expressed uniquely in the form cv + w⊥, where c ∈ R and w⊥ is perpendicular to v. Since Rv(v)= v ✗ v = 0, the "interesting part" of Rv is what it does to vectors orthogonal to v. The set of these vectors is a two-dimensional subspace of R3, the orthogonal complement V⊥ of the 1-dimensional subspace {all multiples of v}. For every w ∈ V⊥, Rv rotates w by π/2 within the plane V⊥, and multiplies the length by ||v||. The sense of the rotation is counterclockwise as seen from the tip of v; i.e. for every nonzero w ∈ V⊥, the ordered triple {w, Rv(w), v} is a right-handed triple of mutually orthogonal vectors. For reasons a little beyond the scope of this course, the linear map Rv is called an infinitesimal rotation. The set of equations in problem 5 says that for all s in the domain of β at which the Frenet frame {T(s), N(s), B(s)} is defined (those s for which κ(s) > 0 ), the derivative of each element of the Frenet frame is given by applying the infinitesimal rotation RA(s) to the vector part of that element.F 2/17/12 2.3/ 6,9,10 M 2/20/12 Let β: I → R3 be a unit-speed curve, let λ be a positive real number, and define a curve γ: I → R3 by γ(t)=λβ(t).
The Curves parametrized by β and γ are similar in the sense of Euclidean geometry: one is simply a "rescaled" version of the other. (In the case of closed Curves, the two curves have different size [unless λ=1] but the same shape.)
- Find an arclength reparametrization μ: J → R3 of γ, where J is a conveniently chosen interval.
- Assume that the curvature function κβ: I → R of β is everywhere positive, so that the torsion function τβ: I → R is defined. Show that the curvature κμ: J → R is also everywhere positive, and find the precise relation between the curvature functions κμ and κβ, and between the torsion functions τμ and τβ. Also find the relation between the function κμ/τμ and the function κβ/τβ.
2.4/ 11ab. (Part (c) is optional; students who know how to use mathematical software packages with 3D graphing capability may enjoy it.) (Continuation of 2.4/11.) I don't know how to get a "hat" over a letter in HTML, so the quantity denoted "a-hat" in 2.4/11, I'll denote a♦ here. On paper, use "a-hat" instead.
- Compute the curvature and torsion of the helix βa♦b (the central curve of the helix βab) in terms of the curvature and torsion of the helix βab.
- What is the relation between the ratio τ/κ for the helix βab, and the ratio τ/κ for its central curve βa♦b?
- Show that there is some λ > 0 such that the helix λ βa♦b has curvature equal to the torsion of βab, and torsion equal to the curvature of βab. (Here λ βa♦b is the curve defined by (λ βa♦b)(t) = λ βa♦b(t).)
W 2/22/12 Redo portions of the previous assignment, as follows:
- For the first problem, change the assumption on β to, "Let β: I → R3 be a regular curve." Then, directly compute the curvature κγ : I → R and (wherever κγ≠ 0) the torsion τγ : I → R of the curve γ in terms of κβ and τβ, without reparametrizing γ.
- 2.4/ 11ab. Redo these without reparametrizing any curves. You should find this somewhat simpler than the way you did these problems the first time. You won't find the simplification enormous, simply because helices with their usual parametrizations are already such computationally friendly curves.
2.4/ 2-5, 17abc. In the definition of "total curvature", if the interval I is unbounded (as in 17abc, where I = R), and/or if the curvature function is unbounded on I, the integral defining the total curvature is treated as an improper integral (which may or may not converge). If this integral diverges, the total curvature does not exist. However, if the integral unambiguously diverges to ∞, we may make a sharper statement than "the total curvature does not exist" by saying that "the total curvature is infinite." F 2/24/12 2.4/ 17d. When you write down the integral for the total curvature, an antiderivative for the integrand will probably not jump out at you, but if you read problem 18, you'll see what the total curvature has to work out to. I want you to do this problem without invoking the general result of #18. Here's one way to proceed.
- Write down the integral that gives the total curvature. The integral will be of the form ∫02π (stuff) dt. Show that this integral can be rewritten as 4∫0π/2 (same stuff) dt.
- Make the magical substitution t = tan–1(– (b/a) cot θ), –π/2 < θ < 0, and see that the integral reduces to something incredibly simple. Then do the integral and find the total curvature. The substitution works just as well without the minus sign, but the sign is related to the next part of this problem.
- Review problem 2.3/8, and try to figure out where the magical substitution above came from (which will tell you why it was destined to work). Note: As you've already seen (if you've read carefully), for plane curves (i.e. curves in R2), when O'Neill writes "N", he means "the vector obtained by rotating T counterclockwise by π/2". This N sometimes equals the principal unit normal, and is sometimes the negative of it. (But unlike the principal unit normal, this N is defined at all points of a regular curve, not just at points at which the curvature is nonzero.) If the curve has any inflection points, the relation between N and the principal unit normal "toggles" each time we pass through an inflection point. When N equals the principal unit normal, we have κ-tilde = κ; when N equals negative of the principal unit normal, we have κ-tilde = – κ.
In class, I use "N" always with the meaning "principal unit normal". Because of the sign-relations above, wherever the curvature of a plane curve in is nonzero,
(κ-tilde)–1(O'Neill's N) = κ–1(my N).
Thus O'Neill's definition of "evolute" that you'll see in 2.4/13 is equivalent to mine, even though you see the notation "(κ-tilde) N" in it.
2.4/ 14. This is a hand-in problem, to be handed in no later than Wed. Feb. 29, at the start of class. You are allowed to use a graphing calculator or graphing software, if you know how. In part (a) (which is a particular case of an example I gave in class, but which I didn't work out all details of), indicate with arrows the direction in which the ellipse is traced out, and the direction in which the evolute is traced out. In part (b), use the following cycloid instead of the one in the book: α(t) = (t – sin t, 1 – cos t), 0 ≤ t ≤ 6π. There is nothing wrong with the book's cycloid; I just want you to use the formula that corresponds to the geometric description that I gave in class and that you have probably seen before.
2.4/ 16. This is a hand-in problem, to be handed in no later than Wed. Feb. 29, at the start of class. First read the motivation for this problem, which is the paragraph directly above the problem. This example is related to the "walking along the edges of a cube" example that I gave in class, in which the torsion was zero wherever it was defined, but the curve was not planar. It shows that this phenomenon can occur even if the curve has no straight segments.
2.4/ 18. Let me add to the book's hint: remember that the speed ||γ'(t)|| of a curve γ is the same as dsγ /dt, where sγ is an arclength function for γ. Apply this to the spherical image. An alternative way to do this problem is to use 2.3/8.
This is a hand-in problem, to be handed in no later than Wed. Feb. 29, at the start of class. Let a > b > 0, and consider the ellipse parametrized by α(t) = (a cos t, b sin t), 0 ≤ t < 2π.
- Find a formula for the evolute of this curve. (Simplify your formular as much as you can.) If your formula is correct, you should see the following expressions in it: a2 – b2, sin3 t, cos3 t.
- [Typo alert: When I originally assigned this problem, I inadvertently reversed the "a2 > 2b2" and "a2 > 2b2" cases. I have just corrected this (3/15/12). For purposes of computing homework grades, I'm treating this problem as if it I had never asked you to hand it in.] Show that (i) if a2 > 2b2, then the evolute lies entirely inside the ellipse; (ii) if a2 = 2b2, then the evolute lies inside the ellipse except for touching the ellipse at (0,b) and (0, –b); and (iii) if a2 < 2b2, then a portion of the evolute lies inside the ellipse, and a portion lies outside. (Remember that we are assuming a > b.)
M 2/27/12 In class we proved that a regular curve with constant nonzero curvature and constant torsion is (part of) a standard, circular helix. Go over the proof and fill in any missing steps or anything that you did not follow in class. For the helix α with equation
α(t) = a(cos(t)e1 + sin(t)e2) + bte3 +d,
where {e1, e2, e3} is an orthonormal basis of R3 and b ≠ 0, determine how the following are related to each other:(You should find, among other things, that any two of these determine the third, but no single one of these determines the other two.)
- The "handedness" (left or right) of the orthonormal basis {e1, e2, e3}.
- The sign of b.
- The sign of the torsion τ.
W 2/29/12 3.1/ 1-4,6,7. (Notes for #7: (i) In between problems 6 and 7, the definition of a {\em group} is given. Those of you who've taken MAS 4301 will already know this definition. (ii) It is more common to call E(3) the Euclidean group in dimension 3 rather than of order 3. "Order of a group" is usually used only for finite groups, where it means the number of elements in the group.) 3.3/ 4. [Typo alert: I originally mis-typed this as 3.4/ 4. I have just corrected it (3/15/12).] The hint given in the book, which is that C has an eigenvector with eigenvalue 1 (equivalently, that the matrix A of C, with respect to a basis, has an eigenvector with eigenvalue 1), needs some justification. Here is an outline of an argument whose details you should fill in. Hint for the remainder of this problem: show that if e is an eigenvector of an orthogonal transformation C, then C preserves the space of all vectors perpendicular to e (i.e. if v⊥e, then C(v)⊥e), a two-dimensional subspace (the orthogonal complement of the span of e). Then apply this fact to a basis {e1, e2} of this orthogonal complement.
- A cubic polynomial with real coefficients has at least one real root. (Hint: if the variable in the polynomial is λ, consider what happens as λ → ∞ and as λ → –∞, and use the Intermediate Value Theorem.)
- If λ3 is a real root of the real, cubic polynomial p(λ), then p(λ)/(λ–λ3) is a quadratic polynomial q(λ) with real coefficients.
- If a quadratic polynomial q(λ) with real coefficients has no real roots, then its roots are a complex-conjugate pair a+ bi, a–bi, where b≠ 0.
- Conclude from the above that if p(λ) is a cubic polynomial with real coefficients, then
p(λ) = c(λ–λ1) (λ–λ2) (λ–λ3),
where c∈R is nonzero, λ3 ∈R, and λ1, λ2 are either both real, or are complex conjugates of each other, say a± bi (with b≠ 0).
- Apply the preceding the to the characteristic polynomial of a 3x3 real matrix A, i.e. the polynomial pA(λ) = det(A–λ I), to show that pA(λ) = –(λ–λ1) (λ–λ2) (λ–λ3), where λ1, and λ2, and λ3 are as above. Recall that λ1, and λ2, and λ3 are the eigenvalues of A. Hence A has at least one real eigenvalue λ3.
- Recall that det(A)= λ1 λ2 λ3. Hence if A is invertible, which is the case for all orthogonal matrices, then it has no zero eigenvalues, so every real eigenvalue is either positive or negative.
- If A, as above, has a pair of complex-conjugate eigenvalues a± bi, deduce that det(A) = (a2 + b2) λ3, and hence that the sign of det(A) is the same as the sign of λ3. Deduce that (in this case), if det(A) > 0 then λ3 > 0.
- Since an orthogonal transformations preserve norms, and since there is at least one eigenvector for every real eigenvalue, the only possible real eigenvalues of an orthogonal matrix are ±1.
- If A is the matrix of an orthogonal transformation of R3 and det(A) > 0, then no matter how many real eigenvalues A has, at least one of the eigenvalues must be 1 (and there must be an eigenvector with this eigenvalue).
F 3/2/12 3.1/ 8,9 3.2/ 1-3 3.3/ 1,5 Recall that an orthogonal transformation S : R3 → R3 is a reflection if there exists an orthonormal basis {e1, e2, e3} of R3 for which S(e1) = e1, S(e2) = e2, and S(e3) = –e3.
- If S is a reflection, what is the inverse map S–1?
- Show that every orthogonal transformation C of R3 of negative determinant is the composition of a reflection and a rotation: C = SR where S is a reflection and R is a rotation. (Hint: use problem 3.4/ 4. You may also get some ideas from the statement of the next problem-part below.)
- Show that the decomposition "C = SR" above is highly non-unique for a given C: for every reflection S, there is a rotation R such that C = SR.
M 3/12/12 It's okay if you don't have this assignment done by Monday's class. Enjoy your spring break. Read the remainder of Section 3.5 (the material after the proof of Theorem 5.3) 3.5/ 3,5,6b W 3/14/12 4.1/ 1,3,4,6-9. In 7a, "the equations ... can be solved for u and v" means that "there exists a pair (u,v) that satisfies the equations," not that there's a mechanical procedure that will produce such a pair (u,v).
If you've taken complex analysis, the function f in #6 may look familiar to you; it's the imaginary part of –(x + iy)3. Getting rid of the minus sign has the same effect as rotating the surface by π about the z-axis; it doesn't change the shape. Similarly, using the real part of (x + iy)3 instead of the imaginary part has the same effect as rotating the surface by π/2 about the z-axis.F 3/16/12 See the typo alert in the assignment for 2/29/12. Do the problem I intended to assign, 3.3/ 4. Other than this, there is no new homework due Friday. Catch up on any old homework you haven't done yet.
I've also fixed a mistake in the last hand-in problem in the assignment for 2/24/12. I'm not asking you to re-do this. None of the methods students used had any chance of showing that the evolute of an ellipse can lie entirely inside the ellipse, no matter what conditions on a and b had been given. When I have a chance, I will write up a solution. If you figure out how to do the corrected problem before I hand out a solution, you may hand in your solution for extra credit. M 3/19/12 4.1/ 10,11,12 W 3/21/12 Read Section 4.2, so you can do the exercises below. I want to forge ahead with Section 4.3 in class on Wednesday, without spending much more time (if any) on 4.2, but I don't want you to miss Examples 2.2 (p. 140) and 2.5 (p. 144), or Definition 2.6 (p. 145). The latter defines ruled surfaces, which are another nice source of examples. 4.2/ 1-5,9. F 3/23/12 4.3/ 1,2,4 M 3/26/12 4.3/ 3 ("Jacobian" here means "determinant of the Jacobian matrix"), 5, 6, 11ab, 13-15 (first read the sentence just above #13) W 3/28/12 Read Section 2.5, "Covariant Derivatives" (of vector fields on R3). We're going to be using the same ideas for vector fields on surfaces shortly, and I will define what's needed just as if Section 2.5 didn't exist, but what we're doing on surfaces will be easier to understand if you read this section first. For the sake of efficiency, I don't want to spend classroom time on Section 2.5 and then redo many of the same things in another lecture on surfaces. 2.5/ 1,2,5 4.3/ 7 (assume Lemma 3.8, an "enhanced" version of which I'll finish in class on Wednesday), 9,10 Past homework to write up for hand-in on Monday, Apr. 2:
- 4.1/ 1, 4
- 4.2/ 2, 5abc (you may hand in 5d for extra credit)
F 3/30/12 We have not covered Sections 4.4 - 4.8 yet, but we've touched on some of the material in these sections. The problems below are doable using only what we have covered.
4.7/ 9 4.8/ 8,9 Towards the end of class on Wednesday, I started Section 5.1; that's what we'll be covering next.
M 4/2/12 5.1/ 1-5. Regarding #s 4 and 5: we have not yet discussed maps from one surface to another, but the definition is self-evident, and that's all that's needed for these problems. (The only thing that may not be self-evident is what "differentiable" means for such a map—eventually, we will adopt our usual convention that "map" means "differentiable map" unless otherwise specified—but that's not needed for these problems.) W 4/4/12 5.1/ 6,9 5.2/ 2,3 F 4/6/12 Finish reading Section 5.2, and start reading Section 5.3. 5.2/ 1,4 M 4/9/12 Finish reading Section 5.3, and start reading Section 5.4. W 4/11/12 5.3/ 1,6,7a F 4/13/12 5.3/ 3,7 5.4/ 1-3, 5, 6, 7. In #5, α(t) = (a1(t), a2(t)). #7 should say "Find the Gaussian curvature ...". In #6 and #7, you can use the formulas derived in #3; just replace (u,v) with (x,y).
5.4/ 3 is an introduction to the subject of geometric partial differential equations. Suppose we ask the question: find a flat surface, or a minimal surface, subject to some other conditions. (Without other conditions, a plane would be a cheap answer.) We can start by looking at surfaces that are given as a graph of a real-valued function f; that's exactly what a Monge patch gives you. The geometric condition "Gaussian curvature identically zero" or "mean curvature identically zero" then translates into a nonlinear partial differential equation for f, which one can try to solve (subject to whatever other conditions are in the problem). Usually, it is extremely difficult to find any closed-form solutions to nonlinear PDEs. Even the existence/uniqueness theory for solutions of nonlinear PDEs is quite challenging. But often the geometric source of a geometric PDE provides insights that one can use to make clever guesses or simplifications. The mathematical literature on minimal surfaces alone is vast.
Generalizations of the geometric PDEs in 5.4/ 3 arise from looking for surfaces of (nonzero) constant Gaussian curvature or constant mean curvature.M 4/16/12 5.4/ 13a, 17 5.7/ 1,3 W 4/18/12 Read Section 4.4. I'll finish going over this on Wednesday, but it will go faster if you do the reading first. Some of the homework problems below will depend on your reading, but most require only what we got through in Monday's class. 4.4/ 3,4ac,5,7,8.
In #5, to understand the idea, replace M by the unit circle in the xy plane. An connected open subset of the circle is either the empty set, the whole circle, or a "proper open arc", an arc that is not the whole circle and doesn't contain its endpoints. On a proper open arc, there is a well-defined, continuous (in fact differentiable) real-valued angle-function θ (a polar-coordinate function). But there is no such function whose domain is the whole circle; if you walk once around the circle counterclockise, extending θ continuously as you walk, then when you get back to your starting point, θ will want to be 2π more than it was when you left home. However, you can cover the circle with three proper open arcs O1, O2, O3, such that each intersection Oi∩Oj is connected (this is the only reason for using three arcs instead of two; it's not important except for the purpose of making a stricter analogy to problem 4.4/ 5), and define a differentiable real-valued angle-function θi on Oi. On each intersection Oi∩Oj, the function θi – θj is constant. The 1-forms dθi and dθj are identically equal on Oi∩Oj. Therefore we can unambiguously a 1-form φ on the circle by setting φ = dθi on Oi; at points in Oi∩Oj, we have dθi = dθj so there is no ambiguity.F 4/20/12 4.4/ 4b Derive all the properties of d listed at the end of class that are not proven in the book. Some of these are parts of 4.4/ 7.
Read Sections 4.5 and 4.6 so that I can cover them more quickly in class. General information My plan for the remaining three lectures is to get through key portions of Sections 4.5, 4.6, 4.7, 6.1, 6.2, 6.5, 6.7, 6.8, and 7.6. Obviously there is far more than three lectures' worth of material in nine sections; even restricting to what I'm thinking of as "key portions", it's doubtful that I'll get through everything I'd like. So the more of this you can read on your own, the better, for those of you who'd like to learn what I'd hoped to get through in this course. M 4/23/12 4.5/ 5,8,10 4.6/ 1,4,5,7,8,9. In #1, α(t) = (α1(t), α2(t)). Re-read Definition 6.1 before doing this problem. For #4, re-read Definition 4.8, p. 164, and the paragraph that follows it. W 4/25/12 4.6/ 10-13