Advanced Calculus Exercises

Advanced Calculus Exercises 2019-2020

Course Textbook: Principle of Mathematical Analysis, 3rd edition, by Walter Rudin

Exercises 1 (p. 21): Do 1, 2, 3, 4, 5

Exercises 2 (p. 22): Do 6, 12, 13, 14

Exercises 3 (p. 43): Do 6, 12, 13, 14

Exercises 4 (p. 44-46): Do 10, 11, 16, 19, 22

Exercises 5 (p. 78): Do 3, 4, 5, 6, 7

Exercises 6 (p. 79-82): Do 9, 14 (a) (b), 16 (a)

Read Theorem 3.20 (p. 57-58), Theorem 3.43-3.44 (p. 70-71) and Examples 3.35 (p.67)

Exercises 7 (p. 98-102): Do 1, 2, 3, 4, 5, 6

Exercises 8 (p. 98-102): Do 8, 9, 13, 20, 21, 22

Exercises 9 (p. 114-119): Do 1, 2, 3, 4, 5, 6

Exercises 10 (p. 114-119): Do 22 (a), (b), (c); 25 (a), (b), (c); 26, 27

Exercises 11 (p. 165-171): Do 1, 2, 4, 7, 15, 16

Exercises 12 Chapter 6 (p. 138-142): Do 1, 2, 4, 5, 10 (a), (b), (c), 11, 15;

Chapter 7 (p. 167): Do 12, 18, 20

Exercises 13 Chapter 8 (p. 196-203): Do 1, 7, 12, 13, 14, 15

Exercises 14 Chapter 9 (p. 239-244): Do 1, 3, 5, 6, 7, 8, 9

Exercises 15 Chapter 9 (p. 239-244): Do 16, 17, 18, 19, 23, 27

Reference: The Elements of Real Analysis 2nd edition, by Robert G. Bartle

Advanced Calculus Exercises (Continued) 2019-2020

Exercises 16 Section 44 (p. 432): Do A, B, C, D, E, F, G

44.A Let A ⊆ R^p, then a point is a boundary point of A if and only if it is a boundary point of the complement C (A) of A. Hence b(A) = b(C (A)).

44.B Let A ⊆ R^p, and let b(A) be the boundary of A.

(a) The set b(A) is closed in R^p.

(b) The interior A^o = A \ b(A) is open in R^p and contains every open set G with G ⊆ A.

(c) The closure ¯A = A ∪ b(A) is closed in R^p and is contained in every closed set F with A ⊆ F.

44.C Let A ⊆ R^p, and let ¯A = A ∪ b(A) be the closure of A. Show that b( ¯A) ⊆ b(A). Give an example to show that equality can hold, and an example the equality can fail.

44.D Let A, B be subsets of R^p. Show that the boundary of each of the sets A ∩ B, A \ B, A ∪ B

is contained in b(A) ∪ b(B). (Hint: b(A) = ¯A ∩C (A).)

44.E A set A ⊆ R^p is closed in R^p if and only if b(A) ⊆ A. A set B ⊆ R^p is open in R^p if and only if B ∩ b(B) = ∅.

44.F If A ⊆ R^p is a bounded subset whose boundary b(A) has content zero, show that boundaries b(A^o) and b( ¯A) of its interior A^o = A \ b(A) and its closure ¯A = A ∪ b(A), respectively, also have content zero and that c(A^o) = c(A) = c( ¯A).

44.G If A ⊆ R^p is a bounded subset whose boundary b(A) has content zero and c(A) > 0, prove that there exists a closed cell K ⊆ A such that c(K) 6= 0.

Exercises 17 Section 45 (p. 451-454): Do A, B, C, D, E, F, G, N

45.A Let Ω ⊆ R^p be an open set and let f : Ω → R^q satisfy a Lipschitz condition on Ω; that is, for some M > 0, kf (x) − f (y)k| ≤ M kx − yk for all x, y ∈ Ω. If K ⊆ Ω is a cube with side length s > 0, show that f (K) is contained in a cube with side length M√

ps. Show that if A ⊆ Ω is a compact set with content zero, then f (A) has content zero, and if B ⊆ Ω is a compact set with content, then f (B) has content.

45.B Consider the polar coordinate map (x, y) = ϕ(r, θ) = (r cos θ, r sin θ) defined on R². If A = [0, 1] × [0, 2π], show that ϕ(A) = D = {(x, y) | x² + y² ≤ 1} and b(D) is the image under ϕ of only one side of A, and that the other three sides of A get mapped into the interior of D.

45.C Consider the polar coordinate map (x, y) = ψ(u, v) = (sin u, sin v) defined on R². Determine the image of the boundary of B = −³₄π,³₄π × −³₄π,³₄π under ψ, and the boundary of ψ(B). Show that most, though not quite all, of the boundary points of ψ(B) are images of interior points of B.

45.D Given that the area of the circular disk {(x, y) | x² + y² ≤ 1} is equal to π, find the areas of the elliptical disks given by

(a)



(x, y) | x² 4 + y²

9 ≤ 1



;

(b) (x, y) | 2x²+ 2xy + 5y² ≤ 1 . (Hint: 2x²+ 2xy + 5y² = (x + 2y)² + (x − y)².)

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Advanced Calculus Exercises (Continued) 2019-2020 45.E Let B be the set {(x, y) | 0 ≤ x, 0 ≤ y, 1 ≤ x + y ≤ 2}. Let u = x + y, v = y so that B is the image under the map (x, y) = ϕ(u, v) = (u − v, v) of the trapezoid C = {(u, v) | 1 ≤ u ≤ 2, 0 ≤ v ≤ u}. Show that ϕ is injective on all of R² and that J_ϕ(u, v) = 1. Deduce that

Z Z

B

(x + y) d(x, y) = Z Z

C

u d(u, v) = 7 3.

45.F Let B = {(u, v) | 0 ≤ u + v ≤ 2, 0 ≤ v − u ≤ 2}. By using the transformation (x, y) 7→

(u, v) = (x − y, x + y), evaluate the integral Z Z

B

(v²− u²) e^(u2+v2)/2d(u, v).

45.G Evaluate the iterated integral

Z 3 1

(Z x²+1 x²

xy dy )

dx

directly. Then use the transformation (x, y) 7→ (u, v) = (x, y − x²) to evaluate this integral.

45.N

(a) By changing to polar coordinates, show that Z Z

CR

e^−(x2+y2)d(x, y) = π

4(1 − e^−R2), where C_R= {(x, y) | 0 ≤ x, 0 ≤ y, x²+ y² ≤ R²}.

(b) If B_L= {(x, y) | 0 ≤ x ≤ L, 0 ≤ y ≤ L}, show that Z Z

BL

e^−(x2+y2)d(x, y) =

Z L 0

e^−x2dx

² .

(c) From the fact that C_R ⊆ B_R⊆ C_R^√₂, show that

R→∞lim

Z R 0

e^−x2dx

²

= π 4, whence it follows that

Z ∞ 0

e^−x2dx = 1 2

√π.

Reference: Calculus on Manifolds, by Michael Spivak

Exercises 18: Do the following from Spivak’s book.

3.9 (a) Show that an unbounded set cannot have content 0.

3.9 (b) Give an example of a closed set of measure 0 which does not have content 0.

3.10 (a) If C is a set of content 0, show that the boundary of C has content 0.

3.10 (b) Give an example of a bounded set C of measure 0 such that the boundary of C does not have measure 0.

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Advanced Calculus Exercises (Continued) 2019-2020

3.18 If f : A → R is nonnegative and Z

A

f = 0, show that {x ∈ A | f (x) 6= 0} has measure 0.

Hint: Prove that {x ∈ A | f (x) > 1/n} has content 0.

4.1 (a) Let {e_i}ⁿ_i=1be the usual basis of Rⁿ and let {ϕ_i}ⁿ_i=1 be the dual basis. Show that ϕ_i1∧ · · · ∧ ϕ_ik(e_i1, . . . , e_ik) = 1.

4.1 (b) Show that ϕ_i1 ∧ · · · ∧ ϕ_ik(v_i1, . . . , v_ik) is the determinant of the k × k minor of





 v₁

... v_k





 obtained by selecting columns i₁, . . . , i_k.

4.2 If f : V → V is a linear transformation and dim V = n, then f^∗ : Λⁿ(V ) → Λⁿ(V ) must be multiplication by some constant c. Show that c = det f.

4.3 If w ∈ Λⁿ(V ) is the volume element determined by T and µ, and w₁, . . . , w_n, show that w(w₁, . . . , w_n) =

q

det (g_ij),

where g_ij = T (w_i, w_j). Hint: If {v_i}ⁿ_i=1 is an orthonormal basis and w_i =

n

X

j=1

a_ijv_j, show

that g_ij =

n

X

k=1

a_ika_kj.

4.4 If w is the volume element determined by T and µ, and f : Rⁿ → V is an isomorphism such that f^∗T = h , i and such that [f (e₁), . . . , f (e_n)] = µ, show that f^∗w = det .

4.20 Let f : U → Rⁿ be a differentiable function with a differentiable inverse f⁻¹ : f (U ) → U. If every closed form on U is exact, show that the same is true for f (U ). Hint: If dw = 0 and f^∗w = dη, consider (f⁻¹)^∗η.

4.23 For R > 0 and n an integer, define the singular 1-cube c_R,n : [0, 1] → R²\{(0, 0)} by c_R,n(t) = (R cos 2πnt, R sin 2πnt). Show that there is a singular 2-cube c : [0, 1]² → R²\ {(0, 0)} such that c_1,n− c_2,n= ∂c.

4.26 Show that Z

cR,n

dθ = 2πn, and use Stokes’ theorem to conclude that c_R,n 6= ∂c for any 2-chain c in R² \ {(0, 0)}.

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