{"id":1889,"date":"2013-12-04T17:56:43","date_gmt":"2013-12-04T22:56:43","guid":{"rendered":"https:\/\/people.clas.ufl.edu\/groisser\/?page_id=1889"},"modified":"2026-03-19T08:13:08","modified_gmt":"2026-03-19T12:13:08","slug":"hw","status":"publish","type":"page","link":"https:\/\/people.clas.ufl.edu\/groisser\/classes\/4930_s14\/hw\/","title":{"rendered":"Homework Assignments"},"content":{"rendered":"\r\n
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Homework Assignments<\/h1>\n\n\n\n

MAT 4930, Section 1374
\nCurves and Surfaces
\nSpring 2014<\/h3>\n\n\n\n

Last updated Apr 22 02:29 EDT 2014 <\/i><\/p>\n\n\n\n

Homework problems and due dates<\/b> (not<\/i> the dates the problems are assigned) are listed below.
\nThis list, especially the due dates, will be updated frequently, usually in the late afternoon or evening
\nthe day of class or the next morning. Due dates, and assignments more than one lecture ahead, are
\nestimates; in particular, due dates may be moved either forward or back, and problems not currently
\non the list from a given section may be added later (but prior to their due dates, of course). Note that
\non a given day there may be problems due from more than one section of the book.<\/p>\n\n\n\n\n\n

Exam dates and some miscellaneous items may also appear below.<\/p>\n\n\n\n\n\n

If one day’s assignment seems lighter than average, it’s a good idea to read ahead and start doing
\nthe next assignment (if posted), which may be longer than average.<\/p>\n\n\n\n\n\n

Unless otherwise indicated, problems are from our textbook (O’Neill, Elementary Differential Geometry<\/i>,
\nrevised 2nd edition). It is intentional that some of the problems assigned do not have answers in
\nthe back of the book or solutions in a manual. An important part of learning mathematics
\nis learning how to figure out by yourself whether your answers are correct.<\/b><\/i><\/p>\n\n\n\n\n\n

Read the corresponding section of the book before working the problems. <\/b> The advice below
\nfrom James Stewart’s calculus textbooks is right on the money:<\/p>\n\n\n\n

    Some students start by trying their homework problems and read the text only if they get stuck
    \non an exercise. I suggest that a far better plan is to read and understand a section of the text
    \nbefore attempting the exercises.<\/i><\/ul>\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
    Date due<\/th>\nSection # \/ problem #s<\/th>\n<\/tr>\n
    W 1\/8\/14<\/td>\nNo problems due. Read Section 1.1.<\/td>\n<\/tr>\n
    F 1\/10\/14<\/td>\n\n
  • Sect. 1.2 (p. 11)\/ 2, 3a,e, 5.<\/li>\n
  • Do the non-book homework problems assigned in class. <\/li>\n
  • Read Section 1.2.<\/li><\/td>\n<\/tr>\n
    • M 1\/13\/14<\/td>\n\n
  • Sect. 1.3 (p. 15)\/ 1, 2, 3abcf, 4, 5. <\/li>\n
  • Read Section 1.3, and start reading Section 1.4.<\/li>\n<\/td>\n<\/tr>\n
    • W 1\/15\/14<\/td>\n\n
  • Sect. 1.4 (pp. 22-23)\/ 4, 7. Note for #4<\/i>: In this book, and in most math books other than Calc 1-2-3 and
    \nDifferential Equations textbooks, “log” means natural<\/i> log (the function you’re used to calling “ln”). Note
    \nfor #7<\/i>: “initial velocity” is defined in problem #6 (which we did in class on Monday).\n<\/li>
  • Finish reading Section 1.4.<\/li><\/td>\n<\/tr>\n
    • F 1\/17\/14<\/td>\nNo new homework.<\/td>\n<\/tr>\n
    W 1\/22\/14<\/td>\n\n
  • Read Section 2.1. (We’ll come back to the remaining sections of Chapter 1 later.)\n<\/li>
  • Sect. 2.1 (pp. 50-52)\/1ace, 2, 3 (“frame” = “orthonormal basis”; see p. 45), 11 (ignore the part about
    \n“(df<\/i>)(v<\/b>)” for now). \n<\/li>
  • Read the rules for hand-in homework<\/a>.\n<\/li>
  • Hand in the following problems:<\/b> Sect. 1.2\/3e; Sect. 1.3\/3c,4,5; Sect. 1.4\/7; Sect. 2.1\/3.\n<\/li><\/td>\n<\/tr>\n
    • F 1\/24\/14<\/td>\nSect. 2.1 (pp. 50-52)\/1bd, 4, 5.\n<\/td>\n<\/tr>\n
    M 1\/27\/14<\/td>\nNo new homework.<\/td>\n<\/tr>\n
    W 1\/29\/14<\/td>\n\n
  • Sect. 2.2 (pp. 57-58)\/3, 4, 8, 10, 11 (assume \u03b1 is regular).
    \n\u00a0\u00a0\u00a0Problems 3 and 4 are examples of something I mentioned in class: coefficients are fine-tuned so that
    \n(assuming you make no mistakes), the speed is the square-root of a recognizable square. In #3, see what
    \nhappens if the coefficient of t<\/i> in the last component of \u03b1(t) is changed to anything other than 1. Similarly,
    \nsee what happens if you change the coefficient of any of the three components of \u03b1(t<\/i>) in #4.\n <\/li><\/td>\n<\/tr>\n
    • F 1\/31\/14<\/td>\n\n
  • Sect. 2.2 (pp. 57-58)\/1, 2, 6c.\n<\/li>
  • Review the rule for hand-in homework<\/a> that says, in boldface, “leav[e] enough space for me
    \nto write comments<\/b>.” Look at the homework returned to you on Wednesday, and ask yourself
    \nwhether you followed this rule. Did you leave ample margins on the left side and<\/i> right side and<\/i>
    \ntop and<\/i> bottom of each page? Did you leave a reasonable amount of space between problems?
    \nIf I had had a sentence-long, or several-sentence long, comment to make anywhere<\/i> in your work,
    \ncould I have fit it in close to what I was commenting on?

    Although the class was very good about following the other rules, roughly half the class didn’t
    \nfollow the leave-me-space-for-comments rule. This particular assignment was mostly computational
    \nand comparatively easy, so in most cases I had very few comments, and so it didn’t matter too
    \nmuch–on this<\/i> assignment–whether you left me space for comments. But in the future, please
    \nmake sure to leave me the comment-space I’ve instructed you to leave. <\/p><\/li>

  • If I had any comments on, or took any points off, your solution to Sect. 1.2\/3e, you should review
    \nDefinition 1.2 on p. 4, and the paragraph that starts with “Thus the value …” just after the definition.
    \nMake sure you understand the difference between the coordinates of a given point<\/i> (the numbers
    \np1<\/sub>, p2<\/sub>, p3<\/sub><\/i> in the context of this definition) and the coordinate functions x1<\/sub>, x2<\/sub><\/i> and x3<\/sub><\/i><\/i> (or x, y,<\/i> and z<\/i>)
    \nfrom R<\/b>3<\/sup> to R<\/b>. Otherwise, much greater confusion could result later in the course.<\/li><\/td>\n<\/tr>\n
    • M 2\/3\/14<\/td>\nNo new homework.<\/td>\n<\/tr>\n
    W 2\/5\/14<\/td>\nDo the problem assigned in class: For the helix discussed in class, compute a<\/i> and b<\/i> in terms of
    \n\u03ba<\/i> and \u03c4<\/i>, and rewrite the formula for \u03b1(t<\/i>) and \u03b2<\/i>(s<\/i>) using \u03ba<\/i> and \u03c4<\/i> in place of a<\/i> and b<\/i>.\n<\/td>\n<\/tr>\n
    F 2\/7\/14<\/td>\n\n
  • Sect. 2.1 (p. 52)\/ 12\n<\/li>
  • Sect. 2.3 (pp. 66–69)\/ 2, 3, 5, 10. Note: In any problem in which the binormal B<\/b> and\/or the torsion \u03c4<\/i> appear, the assumption “wherever \u03ba > 0 <\/i>” is implicit.\n


    \u00a0\u00a0\u00a0 Geometric interpretation of #5<\/b>. For every v<\/b> \u2208 R<\/b>3<\/sup>, the map R<\/i>v<\/b><\/sub>: R<\/b>3<\/sup> \u2192 R<\/b>3<\/sup> defined by R<\/i>v<\/b><\/sub>(w<\/b>) = v \u00d7 w <\/b>
    \nis linear. If v<\/b> = 0<\/b> then R<\/i>v<\/b><\/sub> maps every vector w<\/b> to 0<\/b>, of course. If v<\/b> \u2260 0<\/b>, then every vector w<\/b> can be expressed uniquely in the form c<\/i>v<\/b> + w<\/b>\u22a5<\/sub>, where c<\/i> \u2208 R<\/b> and w<\/b>\u22a5<\/sub> is perpendicular to v<\/b>. Since
    \nR<\/i>v<\/b><\/sub>(v<\/b>)= v \u00d7 v <\/b> = 0<\/b>, the “interesting part” of R<\/i>v<\/b><\/sub> is what it does to vectors orthogonal to v<\/b>. The set of these vectors is a two-dimensional subspace of R<\/b>3<\/sup>, the orthogonal complement<\/i> V<\/i>\u22a5<\/sup> of the 1-dimensional subspace {all multiples of v<\/b>}. For every w<\/b>\u2208 V<\/i>\u22a5<\/sup>, R<\/i>v<\/b><\/sub> rotates w<\/b> by \u03c0<\/i>\/2 within the plane V<\/i>\u22a5<\/sup>, and multiplies the length
    \nby ||v<\/b>||. The sense<\/i> of the rotation is counterclockwise as seen from the tip of v<\/b>; i.e. for every nonzero
    \nw<\/b> \u2208 V<\/i>\u22a5<\/sup>, the ordered triple {w<\/b>, R<\/i>v<\/b><\/sub>(w<\/b>), v<\/b>} is a right-handed triple of mutually orthogonal vectors. For reasons a little beyond the scope of this course, the linear map R<\/i>v<\/b><\/sub> is called an infinitesimal rotation<\/i>.
    \n\u00a0\u00a0\u00a0 For p<\/b>\u2208R<\/b>3<\/sup> and vp<\/sub><\/b>\u2208Tp<\/sub>R<\/b>3<\/sup>, we can analogously define the linear map R<\/i>vp<\/sub><\/b><\/sub>: Tp<\/sub>R<\/b>3<\/sup> \u2192 Tp<\/sub>R<\/b>3<\/sup> by
    \nR<\/i>vp<\/sub><\/b><\/sub>(wp<\/sub><\/b>) = (v \u00d7 w<\/b>)p<\/b><\/sub>. The set of equations in problem 5 says that for all s<\/i> in the domain of \u03b2<\/i> at which the
    \nFrenet frame {T<\/b>(s), N<\/b>(s), B<\/b>(s)} is defined (those s<\/i> for which \u03ba(s)<\/i> > 0), the derivative of each element of the
    \nFrenet frame is given by applying the infinitesimal rotation RA(s)<\/sub><\/i> to that element.\n<\/p><\/li><\/td>\n<\/tr>\n

    • M 2\/10\/14 <\/td>\n\n
  • Sect. 2.1 (p. 52)\/ 12\n<\/li>
  • Sect. 2.3 (pp. 66–69)\/ 1, 6, 8, 9. In #8, “rotating through +90o<\/sup>” means “rotating counterclockwise<\/i> through 90o<\/sup>“.\n<\/li><\/td>\n<\/tr>\n
    • W 2\/12\/14<\/td>\n\n
      \n
    1. (a) Let f<\/i> : R<\/b>\u2192R<\/b>, and define \u03b1 : R<\/b>\u2192R<\/b>3<\/sup> by\n

      \u00a0\u00a0\u00a0\u00a0 \u03b1<\/i>(t<\/i>) = (f<\/i>(t<\/i>) cos(t<\/i>), f<\/i>(t<\/i>) sin(t<\/i>), f<\/i>(t<\/i>)).<\/p>

      \nIf f<\/i> is monotone (increasing or decreasing), the Curve parametrized by \u03b1<\/i> can reasonably be called a
      \n“conical helix”. Figure out why.<\/p>

      \n\u00a0\u00a0\u00a0\u00a0For the rest of this problem, f<\/i> and \u03b1<\/i> are as above. <\/p>

      \n(b) For such a curve \u03b1, write down (in terms of f<\/i> ) the integral that gives the arclength
      \nfunction s<\/i> of \u03b1 that is based at t<\/i>=0 and is consistent with the orientation of \u03b1.<\/p>

      \n(c) Show that \u03b1 is regular provided there is no a<\/i>\u2208R<\/b> for which f<\/i>(a<\/i>) = f’ <\/i>(a<\/i>) = 0. <\/p>

      (d) For the case f<\/i>(t<\/i>) = et<\/sup><\/i>, sketch the Curve parametrized by \u03b1. <\/p>

      (e) Again for the case f<\/i>(t<\/i>) = et<\/sup><\/i>, find an explicit formula for s<\/i>(t<\/i>), solve for t<\/i> in terms of s<\/i>, and write
      \ndown the corresponding unit-speed reparametrization \u03b2<\/i> of \u03b1. <\/p>

      (f) What is the domain of \u03b2<\/i> in part (e)? You should find that it is an interval of the form (\u2013 a<\/i>,\u221e), where a<\/i> > 0. What is the value of a<\/i> telling you geometrically?<\/p>\n<\/li>

    2. Let \u03b2: I<\/i> \u2192 R<\/b>3<\/sup> be a unit-speed curve, let \u03bb<\/i> be a positive real number, and define a curve \u03b3: I<\/i> \u2192 R<\/b>3<\/sup> by \u03b3(t)=\u03bb\u03b2(t).<\/i> (That funny-looking letter for the new curve is a lower-case gamma, rendered poorly by WordPress.) The Curves parametrized by \u03b2<\/i> and \u03b3<\/i> are similar<\/i> 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 \u03bb<\/i>=1] but the same shape.)\n

      (a) Find an arclength reparametrization \u03bc: J<\/i>\u2192 R<\/b>3<\/sup> of \u03b3<\/i>, where J<\/i> is a conveniently chosen interval. <\/p>

      (b) Assume that the curvature function \u03ba\u03b2<\/sub>:I<\/i> \u2192 R<\/b> of \u03b2<\/i> is everywhere positive, so that the torsion
      \nfunction \u03c4\u03b2<\/sub>: I<\/i>\u2192 R<\/b> is defined. Show that the curvature \u03ba\u03bc<\/sub>: J<\/i> \u2192 R<\/b> is also everywhere positive, and find
      \nthe precise relation between the curvature functions \u03ba\u03bc<\/sub><\/i> and \u03ba\u03b2<\/sub><\/i>, and between the torsion functions \u03c4\u03bc<\/sub><\/i>
      \nand \u03c4\u03b2<\/sub><\/i>. Also find the relation between the function \u03ba\u03bc<\/sub><\/i>\/\u03c4\u03bc<\/sub><\/i> and the function \u03c4\u03b2<\/sub><\/i>\/\u03ba\u03b2<\/sub>.<\/i>\n<\/p><\/li><\/ol>\n<\/td>\n<\/tr>\n

    F 2\/14\/14<\/td>\n\n
  • Hand in the following problems:<\/b> Sect. 2.1\/ 12; Sect. 2.3\/8; and the non-book problems
    \n1bdf and 2 assigned with due-date 2\/12\/14. (In your write-up, label the non-book problems
    \n“non-book #1” and “non-book #2”.) <\/li><\/td>\n<\/tr>\n
    • M 2\/17\/14<\/td>\nNo new homework<\/td>\n<\/tr>\n
    W 2\/19\/14<\/td>\n\n
  • Sect. 1.7 (pp. 40-41)\/ 1-5, 7. Read the instructions at the start of the exercises to see what map F<\/i>
    \nthe first four problems refer to. <\/li><\/td>\n<\/tr>\n
    • F 2\/21\/14<\/td>\nNo new homework<\/td>\n<\/tr>\n
    M 2\/24\/14<\/td>\nNo new homework<\/td>\n<\/tr>\n
    W 2\/26\/14<\/td>\nNo new homework<\/td>\n<\/tr>\n
    F 2\/28\/14<\/td>\n\n
  • Redo problem 2 of the assignment due 2\/12\/14, as follows: (i) Change the assumption on \u03b2<\/i> to,
    \n“Let \u03b2: I<\/i> \u2192 R<\/b>3<\/sup> be a regular curve.” (ii) Directly compute the curvature \u03ba\u03b3<\/sub><\/i> : I<\/i> \u2192 R<\/b> and (wherever
    \n\u03ba\u03b3<\/sub><\/i>\u2260 0) the torsion \u03c4\u03b3<\/sub><\/i>and the ratio \u03c4\u03b3<\/sub>\/\u03ba\u03b3<\/sub><\/i> :I<\/i> \u2192 R<\/b> of the curve \u03b3<\/i> in terms of \u03ba\u03b2<\/sub><\/i> and \u03c4\u03b2<\/sub><\/i>, without
    \nreparametrizing \u03b3<\/i>.

    (Note: the problem due 2\/12\/14 originally had “\u03ba\u03b3<\/sub>\/\u03c4\u03b3<\/sub><\/i>” where you now see “\u03c4\u03b3<\/sub>\/\u03ba\u03b3<\/sub><\/i>“. This problem
    \nalways should have had “\u03ba\u03b3<\/sub>\/\u03c4\u03b3<\/sub><\/i>“, since \u03c4(t)<\/i> could be zero for some or all t\u2208 I<\/i>, while \u03ba(t)<\/i> was
    \nassumed nonzero for all t\u2208 I<\/i>.)<\/p><\/li>

  • Sect. 3.1 (pp. 105-107)\/ 1-3, 7, 8. (We’ve done parts of these in class already.) Notes for #7:
    \n(i) In between problems 6 and 7, the definition of a group<\/i> is given. Those of you who’ve taken
    \nMAS 4301 will already know this definition. (ii) It is more common to call E(3)<\/i> the Euclidean group
    \nin dimension 3<\/i> rather than of order 3<\/i>. The terminology “order of a group” is usually used only
    \nfor finite groups, where it means the number of elements in the group.)<\/li>
  • Sect. 3.3 (pp. 115-116)\/ 4. I sketched this in class, but omitted several steps. The hint given in
    \nthe book (see p. 116), which is that C<\/i> has an eigenvector with eigenvalue 1 (equivalently, that the
    \nmatrix A<\/i> of C<\/i>, with respect to a basis, has an eigenvector with eigenvalue 1), needs some
    \njustification. Here is an outline of an argument whose details you should fill in.\n
      \n
    1. A cubic polynomial with real coefficients has at least one real root. (Hint:<\/i> if the variable in the
      \npolynomial is \u03bb<\/i>, consider what happens as \u03bb<\/i> \u2192\u221e and as \u03bb<\/i> \u2192 \u2013\u221e, and use the Intermediate
      \nValue Theorem.)\n<\/li>
    2. If \u03bb<\/i>3<\/sub> is a real root of the real, cubic polynomial p<\/i>(\u03bb<\/i>), then p<\/i>(\u03bb<\/i>)\/(\u03bb<\/i>\u2013\u03bb<\/i>3<\/sub>) is a quadratic polynomial
      \nq<\/i>(\u03bb<\/i>) with real coefficients.\n<\/li>
    3. If a quadratic polynomial q<\/i>(\u03bb<\/i>) with real coefficients has no real roots, then its roots are a
      \ncomplex-conjugate pair a+ bi, a\u2013bi<\/i>, where b\u2260 0<\/i>.\n<\/li>
    4. Conclude from the above that if p<\/i>(\u03bb<\/i>) is a cubic polynomial with real coefficients, then\n

      p<\/i>(\u03bb<\/i>) = c(\u03bb<\/i>\u2013\u03bb<\/i>1<\/sub>) (\u03bb<\/i>\u2013\u03bb<\/i>2<\/sub>)(\u03bb<\/i>\u2013\u03bb<\/i>3<\/sub>), where c<\/i>\u2208R<\/b> is nonzero, \u03bb<\/i>3<\/sub> \u2208R<\/b>, and \u03bb<\/i>1<\/sub>, \u03bb<\/i>2<\/sub> are either both real or are
      \ncomplex conjugates of each other.<\/p>\n<\/li>

    5. Apply the preceding the to the characteristic polynomial of a 3\u00d73 real matrix A<\/i>, i.e. the polynomial
      \npA<\/sub>(\u03bb)<\/i> = det(A\u2013\u03bb I<\/i>), to show that pA<\/sub>(\u03bb)<\/i> = \u2013(\u03bb<\/i>\u2013\u03bb<\/i>1<\/sub>) (\u03bb<\/i>\u2013\u03bb<\/i>2<\/sub>) (\u03bb<\/i>\u2013\u03bb<\/i>3<\/sub>), where \u03bb<\/i>1<\/sub>, and \u03bb<\/i>2<\/sub>, and \u03bb<\/i>3<\/sub> are as above.
      \nRecall that \u03bb<\/i>1<\/sub>, \u03bb<\/i>2<\/sub>, and \u03bb<\/i>3<\/sub> are the eigenvalues of A<\/i>. Hence A<\/i> has at least one real eigenvalue \u03bb<\/i>3<\/sub>.\n<\/li>
    6. Recall that det(A<\/i>)= \u03bb<\/i>1<\/sub> \u03bb<\/i>2<\/sub> \u03bb<\/i>3<\/sub>. Hence if A<\/i> is invertible, which is the case for all orthogonal matrices,
      \nthen it has no zero eigenvalues, so every real eigenvalue is either positive or negative.\n<\/li>
    7. If A<\/i>, as above, has a pair of complex-conjugate eigenvalues a\u00b1 bi<\/i>, deduce that det(A<\/i>) = (a<\/i>2<\/sup> + b<\/i>2<\/sup>) \u03bb<\/i>3<\/sub>,
      \nand hence that the sign of det(A<\/i>) is the same as the sign of \u03bb<\/i>3<\/sub>. Deduce that (in this case), if
      \ndet(A<\/i>) > 0 then \u03bb<\/i>3<\/sub> > 0.\n<\/li>
    8. Since an orthogonal transformations preserve norms, and since there is at least one eigenvector
      \nfor every real eigenvalue, the only possible real eigenvalues of an orthogonal matrix are \u00b11. \n<\/li>
    9. If A<\/i> is the matrix of an orthogonal transformation of R<\/b>3<\/sup> and det(A<\/i>) > 0, then no matter how many real
      \neigenvalues A<\/i> has, at least one of the eigenvalues must be 1 (and there must be an eigenvector with this eigenvalue). \n<\/li><\/ol>\n

      Hint for the remainder of this problem: show that if e<\/b> is an eigenvector of an orthogonal transformation C<\/i>,
      \nthen C<\/i> preserves the space of all vectors perpendicular to e<\/b> (i.e. if v<\/b>\u22a5e<\/b>, then C(v)<\/i>\u22a5e<\/b>), a two-dimensional
      \nsubspace (the orthogonal complement<\/i> of the span of e<\/b>). Then apply this fact to a basis {e<\/b>1<\/sub>, e<\/b>2<\/sub>} of this orthogonal complement.\n <\/p><\/li><\/td>\n<\/tr>\n

    • M 3\/10\/14<\/td>\nIt’s okay if you don’t have this assignment done by Monday’s class. Enjoy your spring break. \n
  • Sect. 3.5\/ 1, 3<\/li><\/td>\n<\/tr>\n
    • W 3\/12\/14<\/td>\n\n
  • 4.1\/ 1,3,4,6-10. In 7a, “the equations … can be solved for u<\/i> and v<\/i>” means that “there exists a pair (u,v<\/i>)
    \nthat satisfies the equations,” not that there’s a mechanical procedure that will produce<\/i> such a pair (u,v<\/i>).


    \u00a0\u00a0\u00a0 If you’ve taken complex analysis, the function f<\/i> in #6 may look familiar to you; it’s the imaginary part
    \nof \u00a0\u2013(x + iy<\/i>)3<\/sup>. Getting rid of the minus sign has the same effect as rotating the surface by \u03c0<\/i> about the
    \nz<\/i>-axis; it doesn’t change the shape. Similarly, using the real part of (x + iy<\/i>)3<\/sup> instead of the imaginary part has the same effect as rotating the surface by \u03c0<\/i>\/2 about the z<\/i>-axis. <\/p><\/li><\/td>\n<\/tr>\n

    • F 3\/14\/14<\/td>\nNo new homework; study for midterm.<\/td>\n<\/tr>\n
    M 3\/17\/14<\/td>\n\n
  • 1.5\/ 1, 3, 4, 5, 6a, 7, 9, 11.<\/li><\/td>\n<\/tr>\n
    • W 3\/19\/14<\/td>\n\n
  • 4.1\/ 10, 11<\/li><\/td>\n<\/tr>\n
    • F 3\/21\/14<\/td>\n\n
  • 4.1\/ 12\n<\/li><\/td>\n<\/tr>\n
    • M 3\/24\/14<\/td>\n\n
  • 4.2\/ 1-4, 9<\/li><\/td>\n<\/tr>\n
    • W 3\/26\/14<\/td>\n\n
  • 4.2\/ 6\n<\/li>
  • 4.3\/ 1, 2, 4<\/li><\/td>\n<\/tr>\n
    • F 3\/28\/14<\/td>\nNo new homework<\/td>\n<\/tr>\n
    M 3\/31\/14<\/td>\n\n
  • 4.3\/ 3bc (see 3a–which we did in class–for context; also be aware that “Jacobian” in 3c means “determinant of the Jacobian matrix”), 5, 6, 11ab <\/li><\/td>\n<\/tr>\n
    • W 4\/2\/14<\/td>\n\n
  • Hand in the following problems:<\/b> Sect. 4.1\/ 4 (prove your answers), 8, 10, 12; Sect. 4.2\/ 9a (the domain D<\/i> is the set of points (u,v<\/i>) in R<\/b>2<\/sup> with -\u03c0\/2 < u <\u03c0\/2<\/i> and no restriction on v<\/i>); Sect. 4.3\/ 3c (in your writeup you may assume the result of 3b), 4b.<\/li><\/td>\n<\/tr>\n
    • F 4\/4\/14<\/td>\n\n
  • Sect. 4.3\/ 7\n<\/li>
  • Sect. 5.1\/ 4 (in part (b), the origin should be excluded from the cone), 5<\/li><\/td>\n<\/tr>\n
    • M 4\/7\/14<\/td>\nNo new homework<\/td>\n<\/tr>\n
    W 4\/9\/14<\/td>\nNo new homework<\/td>\n<\/tr>\n
    F 4\/11\/14<\/td>\nNo new homework<\/td>\n<\/tr>\n
    M 4\/14\/14<\/td>\n\n
  • Sect. 5.1\/ 3\n<\/li>
  • Sect. 5.3\/ 3 (note typo: factor in front of integral should be 1\/(2\u03c0)), 7 (the “canonical isomorphisms” in part (b) are the linear maps Tp<\/sub>R<\/b>3<\/sup> \u2192 Tq<\/sub>R<\/b>3<\/sup> that carry v<\/b>p<\/sub> to v<\/b>q<\/sub>)\n<\/li>
  • Read the examples in Sect. 5.4. See the first three pages of this section for notation. (We covered the material on these pages in class, but not entirely in O’Neill’s notation.)\n<\/li>
  • Sect. 5.4\/ 1,2 <\/li><\/td>\n<\/tr>\n
    • W 4\/16\/14<\/td>\n\n
  • Sect. 5.4\/ 3 (a surface is called flat<\/i> if its Gaussian curvature is identically 0, and minimal<\/i> if its mean
    \ncurvature is identically 0), 6, 7, 13, 17. #7 should say “Find the Gaussian<\/i> curvature …”. In #6 and #7,
    \nyou can use the formulas derived in #3; just replace (u,v<\/i>) with (x,y<\/i>).

    “Minimal surfaces” get their name from the following: Let C<\/i> be a simple closed Curve in R<\/b>3<\/sup>. Consider
    \nsurfaces M<\/i> in R<\/b>3<\/sup> whose boundary is C<\/i>, where “boundary” here means the set of points in R<\/b>3<\/sup> that are not
    \nin M<\/i> but to which some curve in M<\/i> gets arbitrarily close. (For example, the equator of a sphere is the
    \nboundary of the open upper hemisphere.) Among all such surfaces, suppose there is one that has smallest
    \narea. Then the mean curvature of this surface is identically 0. (The proof is beyond the scope of this course.)<\/p>


    Problem 5.4\/ 3 is an introduction to the subject of geometric partial differential equations<\/i>. Suppose we ask
    \nthe question: find a flat surface, or a minimal surface, subject to some other conditions. (Without other
    \nconditions, a plane would be a cheap answer.) We can start by looking at surfaces that are given as a graph
    \nof a real-valued function f<\/i>; that’s exactly what a Monge patch gives you. The geometric condition “Gaussian
    \ncurvature identically zero” or “mean curvature identically zero” then translates into a nonlinear partial
    \ndifferential equation for f<\/i>, 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
    \nexistence\/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.
    \nThe mathematical literature on minimal surfaces alone is vast.<\/p>


    Generalizations of the geometric PDEs in 5.4\/3 arise from looking for surfaces of constant<\/i> (not necessarily
    \nzero) Gaussian curvature or constant mean curvature.\n<\/p><\/li><\/td>\n<\/tr>\n

    • F 4\/18\/14<\/td>\nNo new homework<\/td>\n<\/tr>\n
    M 4\/21\/14<\/td>\n\n
  • Sect. 4.4\/ 1, 4c\n<\/li>
  • Read Sect. 4.5 and Sect. 4.6.<\/li><\/td>\n<\/tr>\n
    • W 4\/23\/14<\/td>\nDo the problems here<\/a>.\n<\/td>\n<\/tr>\n
    <\/td>\n<\/td>\n<\/tr>\n
    <\/td>\n<\/td>\n<\/tr>\n
    <\/td>\n<\/td>\n<\/tr>\n
    <\/td>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\n\n\n



    \n Back to class home page<\/a><\/p>\n\n\n\r\n\t\t\t<\/div>\r\n\t\t<\/div>\r\n\t<\/div>\r\n<\/section>\r\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":245,"featured_media":0,"parent":1936,"menu_order":4,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_acf_changed":false,"featured_post":"","footnotes":"","_links_to":"","_links_to_target":""},"class_list":["post-1889","page","type-page","status-publish","hentry"],"acf":[],"_links":{"self":[{"href":"https:\/\/people.clas.ufl.edu\/groisser\/wp-json\/wp\/v2\/pages\/1889","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/people.clas.ufl.edu\/groisser\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/people.clas.ufl.edu\/groisser\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/people.clas.ufl.edu\/groisser\/wp-json\/wp\/v2\/users\/245"}],"replies":[{"embeddable":true,"href":"https:\/\/people.clas.ufl.edu\/groisser\/wp-json\/wp\/v2\/comments?post=1889"}],"version-history":[{"count":10,"href":"https:\/\/people.clas.ufl.edu\/groisser\/wp-json\/wp\/v2\/pages\/1889\/revisions"}],"predecessor-version":[{"id":5306,"href":"https:\/\/people.clas.ufl.edu\/groisser\/wp-json\/wp\/v2\/pages\/1889\/revisions\/5306"}],"up":[{"embeddable":true,"href":"https:\/\/people.clas.ufl.edu\/groisser\/wp-json\/wp\/v2\/pages\/1936"}],"wp:attachment":[{"href":"https:\/\/people.clas.ufl.edu\/groisser\/wp-json\/wp\/v2\/media?parent=1889"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}