Spring 2007 Math 425 Homework from Spivak's _Calculus on Manifolds_ (due each MONDAY in class as indicated at the left margin - the !ed problems are for you to ponder - these assignments are always subject to change as necessary): [Please read indicated sections even if no problems are assigned] [Supplemental reading or problems from Marsden's & Tromba's _Vector Calculus_ is indicaated by M&T....] [Also, please have a look at my colleague Roman Federov's problems: http://www.math.umass.edu/~fedorov/425/] 2/5 [Warm up with some geometric linear algrebra problems:] An (infinite) chessboard can be thought of as the integer points in the plane: in other words, a possible chess move is of the form mH + nV where H and V are horizontal and vertical moves by a single square, and where m and n are (positive or negative) integers. Analyze the various moves of different chess pieces. For example, can a knight move to any position? How few knight moves would it take to make a basic H or V move? Bishops move diagonally, and thus (unlike the other pieces) they always stay on their same color. These moves are of the form mH + nV where m+n is even; alternatively, they are of the form aA + bB where A = H + V and B = -H + V are basic diagonal moves. Explain why these descriptions of bishops' moves are equivalent (see next problem for one way to go about this). * * * The atoms in salt (NaCl) are located at the points of the integer (cubic) lattice in 3-space: lI + mJ + nK where l, m and n are integers, and where I, J and K are translation vectors along the axes. If we place a Na atom at the origin O, we have seen that there are two descriptions for the sites of the Na atoms: (*) lI + mJ + nK where l+m+n is an EVEN integer, (**) aA + bB + cC where a, b, and c are integers, and where A = I + J, B = J + K, C = I + K are diagonal vectors on the face of the cube. We want to see that these two descriptions of the Na sites (also known as the face-centered cubic, or FCC, lattice) are equivalent. To do this, find a formula F(a, b, c) = (l, m, n) which gives the location in description (*) of a point V = aA+ bB + cC in description (**). Also find an inverse formula G(l, m, n) = (a, b, c) which lets us recover the coefficients in (**) from those in (*). * * * Consider a tetrahedron in R^3 with vertices A, B, C, D. Show that the (outer) face normal vectors (weighted by their respective areas) sum to the zero vector. (Hint: express areas using cross products.) Generalize this to any polyhedron in R^3. (Hint: dissect into tetrahedra.) What generalizations can you make to R^n? (Recall the case n=1 is trivial, and n=2 was done in class, so we are interested in n>3.) (Hint: use determinants.) * * * [please read the Preface and Chapter 1 of Spivak] 2/12 Spivak: 1-1, 1-2, 1-3, 1-4!, 1-5, 1-6, 1-7, 1-8!, 1-9 1-10, 1-11!, 1-12! (Hint: think transpose), 1-13 M&T: 1.5 2/19 Spivak: 1-14, 1-15, 1-16, 1-19, 1-23, 1-24, 1-25, 1-27, 1-28 M&T: 2.2 2/26 Spivak: 2-1, 2-2, 2-6, 2-7!, 2-8! 2-9!, 2-10h,a,b,j,e!, 2-13!, 2-14!, 2-15!, 2-16 M&T: 2.3, 2.5, 2.6 3/5 Spivak: 2-17h,a,e, 2-18, 2-20a,c,e, 2-21!, 2-22!, 2-23, 2-25!, 2-29, 2-32, 2-34!, 2-35! M&T: 3.1, 3.2, 3.3 3/12 Spivak: 2-36!, 2-37!, 2-38, 2-40! Space of circular arcs problem (as discussed in class): please work out the linear approximation of (s,c) in terms of (R,\theta) at other locations besides the semicircle (\theta = \pi - suggestions are near \theta = \pi/k where k = 2, 3...) M&T: 3.4, 3.5 [MIDTERM EXAM in class on Monday 4/2 - please think about the following two "topical" functions: A) (x,y) -> (e^x cos y, e^x sin y): R^2 -> R^2 B) (x,y,z) -> 1 - x^2 - y^2 - z^2: R^3 -> R^1 - think about all the things I could ask you about these...!] 4/9 Recall that a subset S of \R^n has (n-dimensional Lebesgue) measure zero if it can be covered by open rectangles R_1, R_2,..., R_i,... such that for an real e > 0, \sum m(R_i) < e. (Here m(R_i) is n-volume of the rectangle. i.e. the product (b_1-a_1)...(b_n-a_n) if R_i = (a_1, b_1)\times...\times(a_n, b_n). Show that the subset \Q^n of \R^n consisting of all vectors with rational coordinates has measure zero. [Hint: first do the case n=1, recalling that \Q is countable.] Also show that the boudary of a rectangle R = [a_1, b_1]\times... \times[a_n, b_n] has measure zero (and consquently, we could have alternatively used closed rectangles in defining measure zero). [Hint: first show more generally that an (n-1)-plane in \R^n has measure zero.] Spivak: 3-1, 3-5, 3-6, 3-7!, 3-8, 3-10, 3-14, 3-18! 4/16 Spivak: 3-27 [there's a printing error: explain why the missing upper limits on the right are both b], 3-28, 3-35, 3-36, 3-40, 3-41 4/23 Spivak: 4-1, 4-2, 4-3, 4-8, 4-19, 4-20!, 4-21 4/30 Spivak: 4-23, 4-24, 4-25, 4-26, 4-27, 4-28!, 4-31!, 4-33!, 4-34! 5/7 Spivak: 5-1, 5-5, 5-6, 5-9, 5-15!, 5-16, 5-17 5/14 Spivak: 5-18!, 5-20!, 5-22!, 5-25, 5-26, 5-31, 5-32, 5-33.... ===================================================================== Here is your FINAL EXAM, on which I expect you to work ALONE and which I would like to you to complete on or before Saturday 26 May 2007. You may place the exam under my office door (LGRT 1435G) IN A SEALED ENVELOPE, or you may mail it to me (postmarked on or before Saturday 26 May 2007) Rob Kusner 17 May 2007 ===================================================================== 1. Even though polar coordinates (r,t) on the plane are not both globally defined functions (for example, the angle "function" t is multivalued and undefined at the origin) on the plane R^2, the corresponding 1-forms are well defined (at least on the punctured plane R^2 \ {(0,0)}), and can be expressed using cartesian coordinates (x,y). a. Using the relations x = r cos t, and y = r sin t, find functions A, B, C and D which express the 1-forms dr = A(x,y) dx + B(x,y) dy and dt = C(x,y) dx + D(x,y) dy in terms of dx and dy. b. On what open subsets are A, B, C, D continuous? ... differentiable? c. Show that the integral of dr around any closed loop in the punctured plane R^2 \ {(0,0)} is zero. d. Compute the integral of dt around the (counter-clockwise oriented) unit circle c parametrized by c(s) = (cos s, sin s). e. Suppose b is any loop which winds once (counter-clockwise) around the origin. Show that the integral of dt around b is the same as for the unit circle c. (Hint: show d(dt)=0 and use Stokes' Theorem.) f. Generalize part e to a loop which winds n times around the origin. (For example, the circle above winds n=1 times, while the clockwise circle b(s) = (cos s, - sin s) winds n=-1 times, and a loop which doesn't surround the origin winds n=0 times.) 2. Let W=dx*dy*dz (I will let * stand for wedge product for this exam) denote the volume 3-form on R^3, let S be surface (such as a plane or sphere or whatever) in R^3 with unit normal vector field N, and let w = W(N,_,_) be the 2-form that results from inserting N in the first of the three slots of W. a. In case S is the xy-plane {z=0} with upward unit normal N = e_3, show that w = W(N,_,_) = dx*dy, namely the area 2-form on S. b. In case S is the graph of a function f(x,y) over this plane, find a forumla for the upward unit normal in terms of grad(f)=(f_x, f_y), and show that w = W(N,_,_) = (1+|grad(f)|^2)^{1/2} dx*dy is again the area form for the graph. c. When S is the unit sphere around the origin with outer unit normal N, show w = W(N,_,_) = x dy*dz - y dx*dz + z dx*dy, and argue geometrically that is again the area form on S (in fact, your argument should work for any "nice" surface with unit normal N). d. Using the relation x^2+y^2+z^2=1 to express each of the terms in the 2-form w on the appropriate hemisphere and corresponding unit disk (for example z dx*dy = (1 - x^2 - y^2)^{1/2} dx*dy for the upper hemisphere over the unit disk in the xy-plane) integrate w over S. (The 6 integrals should sum to area(S) = 4\pi). e. Thinking of w = x dy*dz - y dx*dz + z dx*dy as a 2-form on R^3, compute the 3-form dw on R^3. f. Apply Stokes' Theorem to the unit ball B with boundary S (the unit sphere) and conclude vol(B) = area(S)/3 = 4\pi/3 Generalize this argument to show that the n-volume of the unit ball B in R^n and the (n-1)-area of its boundary (n-1)-sphere satisfy the relation n vol(B) = area(S). ===================================================================== Do your best with this - good luck!!! ===================================================================== Have a great summer & here's hoping to do math with you in the future!