CAAM 402 Homework Assignments: Spring 2009

"Chapter.section.problem" numbering refers to the text.

Homework should be turned in, in class, on the due date. No late homework accepted, except with prior agreement of the instructor.

Each problem is worth 10 points, unless otherwise noted.

1. Assigned 13 Jan 09, due 20 Jan 09.
1. IX.2.1
2. IX.2.2
3. IX.2.12
4. IX.3.1
5. IX.3.3: parts a, b, d, e, i
6. IX.3.6
7. IX.5.1
8. IX.5.5
9. IX.6.1
10. IX.6.7
2. Assigned 20 Jan 09, due 27 Jan 09
1. IX.2.11
2. IX.2.13
3. IX.2.14
4. IX.3.5
5. IX.5.3
6. IX.6.4 (a special case is lim n^1/n=1)
7. IX.6.5
8. IX.7.1
9. IX.7.5 (Borel's theorem). This result shows just how non-analytic C-infty functions can be - that is, far from having a positive radius of convergence, Taylor series may have arbitrary coefficients!!! First part: show that f with the prescribed properties can be constructed as indicated.
10. IX.7.5 Second part: show that g has the property claimed.
3. Assigned 27 Jan 09, due 3 Feb 09
1. XV 1.1
2. XV 1.2
3. XV 1.12
4. XV 2.1
5. XV.2.2
6. XV 2.3
7. XV 2.4
8. XV 2.5
9. XVII 2.1
10. XVII 2.2
4. Assigned 3 Feb 09, due 10 Feb 09
1. X 6.2
2. XV 2.9
3. XV 2.11
4. (pdf)
5. XVII 4.1
6. (pdf)
7. (pdf)
8. (pdf)
9. XVII 5.1
10. XVII 5.2
5. Assigned 12 Feb 09, due 19 Feb 09 - THIS ASSIGNMENT IS PLEDGED.
1. XVII 6.1, first part (ending with "[Hint: Induction]").
2. XVII 6.1, second part ("Also...").
3. XVIII 1.1 (a)
4. XVIII 1.1 (b)
5. XVIII 1.3
6. XVIII 1.4
7. XVIII 1.5
8. XVIII 2.1(a)
9. XVIII 2.1(b)
10. XVIII 2.1(c)
6. Assigned 23 Feb 09, due 10 Mar 09.
1. XVIII 2.4
2. XVIII 3.1
3. XVIII 3.2
4. XVIII 3.3
5. XVIII 3.4
6. XVIII 5.1
7. XVIII 5.2
8. XVIII 5.3 (example where image of R is not all of R: try f(x)=e^x)
9. XVIII 5.4
10. XVIII 5.5
7. Assigned 10 Mar 09 due 17 Mar 09
1. XVIII 5.6
2. XVIII 5.8
3. XVIII 5.9
4. XVIII 5.12
5. XVIII 5.13
6. XIX 1.1
7. XIX 1.2 (hint: alpha(n+1)-alpha(n) must go to zero as n goes to infinity, since alpha(t) converges to P as t goes to infinity. Express alpha(n+1)-alpha(n) as an integral using the MVT, substitute f(alpha(t)) for d alpha / dt (it solves the DE!) under the integral sign. Note that f(alpha(t)) must converge to f(P), and supply enough reasons to conclude that the integral must converge to f(P) also. That does it.)
8. XIX 3.1
9. XIX 3.3
10. XIX 3.5(a) (the linear map-valued function C(s,t) is usually called the Fundamental Solution or something similar)
8. Assigned 19 Mar 09, due 26 Mar 09 - THIS ASSIGNMENT IS PLEDGED.
1. XVIII 5.7
2. XVIII 5.10
3. XVIII 5.11
4. XVIII 5.14
5. XIX 1.4(a)
6. XIX 1.4(b)
7. XIX 1.4(c)
8. XIX 3.2
9. XIX 3.5(b),(c),(d)
10. XIX 3.6 (This is the justification of Duhamel's principle, or the variation of constants formula, as presented in class. Explain exactly how you differentiate the second term using the chain rule, and why differentiation under the integral sign is justified - quote the relevant theorem from last semester, and show why problem 3.5 justifies the use of this theorem.)
9. Assigned 26 Mar 09, due 2 Apr 09. Use the definition of "integrable function" given in class: an integrable function on a rectangle is a uniform limit of simple functions (i.e. is in the closure of the simple functions in the Banach space of bounded functions). Its integral is the limit of the integrals of any approximating sequence of simple functions, which exists because the integral is sup-norm continuous on the simple functions:
1. Spivak 3-1 (this is REALLY easy if you follow the preceding advice!)
2. Spivak 3-2
3. Spivak 3-4
4. Spivak 3-5
5. Spivak 3-6
6. Spivak 3-8 (or 3-25) - you may use "measure 0" in place of "content 0" - makes no difference by Theorem 3-6. You may find it helpful to note that a rectangle with sides of finite length (like the one in the problem) is compact, so you can fatten up each rectangle in the set with total volume less than epsilon to an open rectange of twice the volume, and these "fattened" rectangles form an open cover. There must be a finite subcover, which has total volume less than 2*epsilon, and the closures of these rectanges form a cover by closed rectangles with the same total volume bound. Now you might consider projecting onto the first n-1 coordinates, and using induction.
7. Spivak 3-10 (b) (hint: what is the boundary of the set of rational points in [0,1]?)
8. Spivak 3-14
9. Spivak 3-35 (a) (the first and third cases are straightforward, since by definition (p. 56) the volume of a rectangle is what it usually means and the images in these cases are rectangles. The second case is definitely nontrivial. You may use Fubini's theorem to make life easier: the image is the cartesian product of a rectangle (product of [a_i,b_i], i anything but j or k), and a 2-dimensional trapezoid. Then do the 2-dimensional case carefully. It may help to consult Lang, pp. 587-8.)
10. Spivak 3-35 (b) (the issue here is justifying the hint, which you must do carefully: do the triangular case first, then use the fact that any matrix can be written as a product of a lower triangular matrix and an upper triangular matrix (LU decomposition)).
10. Assigned 3 April 09, due 9 April 09:
1. Spivak 1-30 (the definition of oscillation is on the bottom of p. 12 and the top of p. 13. Your first step should be to show that osc(f,a)=the difference between the limits of f from right and left, respectively, because f is increasing, or equivalently inf {f(x): x greater than a} - sup {f(x): x less than a}.)
2. Spivak 3-12
3. Spivak 3-25
4. Spivak 3-26 (thus justifying all the nonsense you've heard over the years about the integral being "the area under the curve". In this case it's easiest to use the upper integral = lower integral definition of "integrable": you can find partitions with upper and lower sums as close as you like, and the difference is the area of a bunch of rectangles enclosing the only interesting part of the boundary of A_f.)
5. Spivak 3-27 (in my copy the upper limits of integration on the RHS did not print - they are b and b, respectively)
6. Spivak 3-29
7. Spivak 3-31
8. Spivak 3-32 (of course this is just the theorem on differentiating under the integral sign - the FTC plus Fubini (interchange the order of integrals!) gives a different proof than the one you saw last semester)
9. Spivak 3-41(a),(b)
10. Spivak 3-41(c),(d),(e) (hint: for any r, there's an s > r so that B_r is a subset of C_r is a subset of B_s, and the integrand is positive)
11. Assigned 9 April 09, due 16 April 09.
1. A (closed) cube of diameter in Rⁿ is exactly what it sounds like, that is, a ball in the sup norm of radius d/2. Show that any rectangle R can be covered with a finite collection {B_i,i=1,...N} of cubes for so that the sum of v(B_i) is less than 2 v(R).
2. Show that a set C is Jordan-measurable if and only if for any &epsilon > 0, its boundary is a subset of a union of closed cubes with total volume < &epsilon (that is, you can replace "rectangle" with "cube" in the definition).

   The setup for the next several problems: U is an open set in Rⁿ, K is a compact subset of U, C is a compact, Jordan-measureable (definition: Spivak, p. 56) subset of K^int, g is a C¹ diffeomorphism from U to Rⁿ.

3. Show that there exist L, M with 0 < L &le M so that for any cube B &sub K, there is a cube B' &sub g(U) so that B' &sup g(B) and Lv(B') &le v(B) &le Mv(B').
4. Show by example that for a suitable choice of U, K, and g, there exist rectangles R so that (i) v(R) is as small as you like, and (ii) the volume of the smallest cube containing g(R) is at least 1. So in this sense rectangles and cubes are different. (Hint: R is going to be long and skinny, and g is going to bend it.)
5. Show that g(C) is Jordan-measureable. (This is the point of the last few problems, of course.)
6. Suppose that f is Riemann-integrable on a rectangle R. Show that the absolute value of the integral of f over R is less than or equal to the sup norm of f times the volume of R. [This is total review - it goes by observing that the same holds for simple functions and taking the limit. Write the proof out carefully.]
7. Suppose that f is continuous on g(U). Show that for any &epsilon > 0, there is a continuous f₀ with supp f₀ &sub g(C)^int so that the integral of f₀ differs from the integral of f over g(C) by less than &epsilon. [Hint: g(C) is Jordan-measureable: cover its boundary with closed cubes of total volume less than &epsilon. Make two additional families of cubes by expanding each cube of the first family by a factor of two and three respectively; the total volumes of the second and third families are less than 2ⁿ&epsilon and 3ⁿ&epsilon respectively. The interiors of the third family form an open cover of the boundary &part C, which is compact, so finitely many of them, say {U₁,...,U_N}, still cover. The union of the corresponding finite subset of the second family (of closed cubes) is closed, and its complement in g(C) is an open subset U₀ of g(C)^int. {U₀,U₁,...,U_N} is an open cover of g(C). Choose a partition of unity {&phi₀,...,&phi_N} subordinate to it. Then the function you want is &phi₀f. You can express the integral of f as a sum of integrals &phi_if, and the difference in question is just the sum of these for i=1,...N, which you can estimate by the triangle inequality and problem 6.

   If you follow this hint, carefully write out the proof of every statement in the preceding paragraph!]

8. Use the version of the change-of-variables formula proved in class, together with the results of the preceding exercise, to prove that

   &int_g(C)f = &int_C f &omicron g |det g'|

   [Hint: you will need to keep around the partition of unity from the last problem, and observe that {&phi_i &omicron g} is a partition of unity for C, with similar properties.]

12. Assigned 17 April 09, due 29 April 09 (last day of final exam period) - THIS ASSIGNMENT IS PLEDGED. Please turn it in by 1700 on 29 April 09 (or earlier) by dropping it in the box on the chair outside my office door, in the folder marked "HW 12".
1. Use the version of the change of variables theorem established in the last assignment to prove that the volume of the image A(K) of a rectangle K &sub Rⁿ under a linear map A &isin L(Rⁿ,Rⁿ) is

   v(A(K)) = det(A) v(K)

   [Notes: volume of a Jordan-measureable set is defined on p. 56 in Spivak. In the preceding equation, I have confounded the name of A with the name of its matrix in the standard basis. Note that Spivak proposes an "elementary" proof of this fact in problem 3-35, and uses it in his proof of the C-of-V theorem (3-13). I used one of the three special cases listed in 3-35 in my proof of the C-of-V theorem. The second case is actually a bit involved; Lang works through it in detail. On the other hand the third case, which I used, is trivial. So you get the whole thing from the (trivial) third case and the rest of the proof of the C-of-V formula, which you had to go through anyway.]

2. The cross product x × y in R³ is defined in Spivak problem 4-9 (b) (from which the formulas in 4-9 (a) follow immediately), and pages 83-84 relate it to the determinant. Here is an alternate derivation: for a vector x &isin R³, define the alternating 1-tensor

   &omega¹_x=x₁e₁^* + x₂e₂^* + x₃e₃^*

   and the alternating 2-tensor

   &omega²_x= x₁e₂^* &and e₃^* + x₂e₃^* &and e₁^* + x₃e₁^* &and e₂^*.

   Show that for any x,y &isin R³,

   &omega¹_x &and &omega¹_y = &omega²_{x × y}.

3. Spivak 4-19(a) (recall that a vector field is simply a vector valued function; the notation echoes that of the last problem.)
4. Spivak 4-19(b)

The following problems develop standard integration rules for the disk and circle in R². Similar rules hold for the ball and sphere in R³, involving spherical coordinates.

5. The unit disk D is the set {x &isin R²: |x| &le 1} (the norm is the Euclidean norm - else the boundary is not smooth!!!). The unit circle C is its boundary: C=&part D. Show that D is a submanifold with boundary of R².

   [Hint: you need to exhibit an explicit atlas of charts. In view of the next problem, I suggest a family of atlases depending on a parameter &epsilon > 0, as follows: define open sets

   U₁={x=(x₁,x₂): &epsilon < |x| < 1+&epsilon, -1+&epsilon < x₁/|x|},

   U₂={x=(x₁,x₂): -1 + &epsilon < x₁ < 2 &epsilon, -2 &epsilon < x₂ < 2 &epsilon},

   and

   U₃={x=(x₁,x₂): -1 - 2&epsilon < x₁ < -1+2 &epsilon, -2 &epsilon < x₂ < 2 &epsilon}.

   So U₁ is an annulus with a thin pie-slice cut out of it around the negative x₁ axis, U₂ is a long thin rectangle with area approximately &epsilon, and U₃ is a square of area 4 &epsilon². These overlap and cover D. Now choose for your chart maps

   (1) (essentially) polar coordinates in U₁: c₁(x)=(r(x)-1, &theta(x)), r(x)=|x|, &theta(x)=the unique solution of cos &theta = x₁/|x|, sin &theta = x₂/|x| between -&pi and +&pi (see Spivak problem 3-41 (b) for an explicit description);

   (2) c₂(x)=(x₁-1,x₂) (this is just shifting every point to the left by 1 in the first ("x") direction - note that U₂ is a subset of D^int, and shifting it to the left by 1 puts its image in the interior of the half-space {x₁<1}.);

   (3) c₃(x)=(r(x)-1,-x₂).

   Check that

   (1) each of the c's is a diffeomorphism,

   (2) c_i maps U_i &cap D to H = {x: x₁ &le 0}, i=1,2,3;

   (3) for each i, C &cap U_i = {x &isin D &cap U_i: c_i,1(x)=0}, that is, c_i maps C &cap U_i into &part H &cap c_i(U_i);

   (4) The restrictions of the charts c_i to C make C into an oriented submanifold of R²: oriented means that the overlap maps c_i o c_j^-1 are orientation preserving, i.e. have Jacobian determinant > 0, and this is the only thing you have to check.

   Taking care of (4) is a lot easier if you remember that U₂ doesn't intersect C so you can forget about c₂ and need only check one overlap map, and c₁^-1(r,&theta)=(r cos &theta, r sin &theta). Also note that in the overlap U₁ &cap U₃, &theta is near ± &pi . End of Hint]

6. Let f &isin C^&infin(R²). Show that

   ∫_D f dx₁ &and dx₂ = ∫₀¹ ∫_-&pi^&pi r dr d&theta f(r cos &theta, r sin &theta).

   [Hint: Use the atlas family constructed in the previous paragraph. Split the integral on the LHS into a sum of three using a partition of unity subordinate to the atlas. Bound the integrals coming from U₂ and U₃ by multiples of &epsilon and &epsilon² respectively, and show that the integral over U₁ converges to the RHS as &epsilon &rarr 0. Thus the LHS converges to the RHS as &epsilon &rarr 0. Since both LHS and RHS are actually independent of &epsilon, they must have been equal in the first place.]

7. Let F: R² &rarr R² be a smooth vector field (to remind you, "smooth" means "infinitely often differentiable"), and define &omega¹_F &isin &Omega¹(R²) as in problem 4-19 in Spivak. Show that

   ∫_C &omega¹_F = ∫_-&pi^&pi d&theta (F₂(cos &theta, sin &theta) cos &theta -F₁(cos &theta, sin &theta) sin &theta)

   [Note: you will need to construct an argument similar to the one used in the previous problem.]

8. Establish the formula in Spivak, problem 4-21 (if it helps, use the explicit formulas given in problem 3-41). Use it to show that

   ∫_C x dy - y dx = 2 π.

9. Green's theorem for the circle: use Stoke's theorem (5-5) and the preceding exercises to show that

   ∫_-&pi^&pi d&theta (F₁(cos &theta, sin &theta) cos &theta +F₂(cos &theta, sin &theta) sin &theta) = ∫_D &part F₁/&part x₁ + &part F₂/&part x₂.

10. Suppose that C₁ and C₂ are concentric circles, centered at the origin, with radii r₁ < r₂. Suppose that F is a smooth vector field satisfying

    &part F₁/&part x₁ + &part F₂/&part x₂ = 0

    at every point (diction: F is "divergence-free"). Denote by n the outward unit normal on either circle, regarded as the boundary of the disk of the same radius. Show that

    ∫_C1 F^T n = ∫_C2 F^T n.

    [Hint: Let A be the annulus bounded by the two circles. The boundary of A is the union of the two circles; on the outer circle, the outward unit normal of A is the same as the outward unit normal of the the disk of the larger radius; on the inner circle, the outward unit normal of A is the negative of the outward unit normal of the disk of the smaller radius. Apply the Divergence Theorem.]