Let A ⊂ Rn be a bounded set and let f : A → R be a bounded function

Integral as an iterated integral

Let U be an open subset of Rⁿ and let f : U → R^m be a map defined on U with values in R^m. 1. Let K = I1× · · · × In, where Ij = [aj, bj] for each j = 1, . . . , n. Remember that

• the content (or volume) c(K) of K is defined to be

c(K) = (b₁− a₁) × · · · × (b_n− a_n) =

n

Y

j=1

(b_j− a_j).

• a set Z ⊂ Rⁿ has content (or volume) zero if ∀  > 0, ∃ a finite collectionC = {Kj}^m_j=1 of n-cells such that

Z ⊂

m

[

j=1

Kj, and

m

X

j=1

c(Kj) < ε.

2. Let A ⊂ Rⁿ be a bounded set and let f : A → R be a bounded function. Suppose that K is a closed cell containing A and let fK : K → R be an extension of f defined by

f_K(x) =

(f (x) if x ∈ A, 0 if x ∈ K \ A.

We say that f is integrable on A if f_K is integrable on K, and define Z

A

f = Z

K

f_K.

3. (The Jacobian Theorem) Let Ω ⊆ R^p be open. Suppose that

φ : Ω → R^p belongs to class C¹(Ω) is injective on Ω, J_φ(x) 6= 0 for x ∈ Ω,

A has content and ¯A ⊂ Ω.

If ε > 0 is given, then there exists r > 0 such that if K is a closed cell with center x ∈ A and side length less than 2r, then

|J_φ(x)|(1 − ε)^p ≤ c(φ(K))

c(K) ≤ |J_φ(x)|(1 + ε)^p. 4. (Change of Variables Theorem) Let Ω ⊆ R^p be open. Suppose that

φ : Ω → R^p belongs to class C¹(Ω) is injective on Ω, J_φ(x) = det dφ(x) 6= 0 for x ∈ Ω,

A has content and ¯A ⊂ Ω and

f : φ(A) → R is bounded and continuous.

Then Z

φ(A)

f = Z

A

(f ◦ φ)|J_φ|.

Partition of unity

1. For each p ∈ Rⁿ, and for any 0 < r₁ < r₂, there exists a continuous (or a smooth) function g : Rⁿ → R such that

g(x) =

(1 if x ∈ B_r1(p), 0 if x /∈ B_r2(p).

2. Suppose K is a compact subset of Rⁿ, and {V_α} is an open cover of K. Then there exist functions ψ₁, . . . , ψ_s∈ C^∞(Rⁿ), the space of smooth functions on Rⁿ, such that

(a) 0 ≤ ψ_i ≤ 1 for 1 ≤ i ≤ s;

(b) each ψ_i has its support in some V_α, i.e. {x ∈ Rⁿ | ψ_i(x) 6= 0} ⊂ V_α, and (c)

s

X

i=1

ψ_i(x) = 1 for every x ∈ K.

Because of (c), {ψ} is called a partition of unity, and (b) is sometimes expressed by saying that {ψ_i} is subordinate to cover {V_α}.

3. If f ∈ C (Rⁿ) is a continuous function in Rⁿ and the support of f lies in K, then f (x) =

s

X

i=1

ψ_i(x)f (x) for x ∈ Rⁿ. Note that each ψ_if has its support in some V_α.

4. Let A ⊂ Rⁿ and let O be a collection of open subsets of Rⁿ covering A. Then there is a collection Φ of continuous functions ϕ defined in an open set containing A, with the following properties:

(a) For each x ∈ A we have 0 ≤ ϕ(x) ≤ 1.

(b) For each x ∈ A there is an open set V containing x such that all but finitely many of ϕ ∈ Φ are 0 on V.

(c) For each x ∈ A we have X

ϕ∈Φ

ϕ(x) = 1 (by (b) for each x this sum is finite in some open set containing x).

(d) For each ϕ ∈ Φ there is an open set U ∈ O such that ϕ = 0 outside of some closed set contained in U.

A collection Φ satisfying (a) to (c) is called a continuous partition of unity for A. If Φ also satisfies (d), it is said to be subordinate to the cover O. In this chapter we will only use continuity of the functions ϕ.

5. Let A ⊆ Rⁿ be an open subset and let O be an open cover of A. Then O is an admissible open cover if every U ∈O is contained in A.

LetO be an admissible open cover of the open set A ⊆ Rⁿ, Φ be a partition of unity for A subordinate to O, and let f : A → R be bounded in some open set around each point of A.

Then we say f is integrable (in the extended sense) on A if X

ϕ∈Φ

Z

A

ϕ · |f | converges.

This implies convergence of X

ϕ∈Φ

| Z

A

ϕ · f |, and hence absolute convergence of X

ϕ∈Φ

Z

A

ϕ · f, which we define to be

Z

A

f, i.e. if X

ϕ∈Φ

Z

A

ϕ · |f | converges, then Z

A

f :=X

ϕ∈Φ

Z

A

ϕ · f.

6. Recall that a subset A of Rⁿ has content 0 if for every ε > 0 there is a finite cover {K₁, . . . , K_m} of A by closed n-cells such that

m

X

j=1

c(K_i) < ε. A subset A of Rⁿ has mea- sure 0 if for every ε > 0 there is a cover {K1, K2, K3, . . .} of A by closed n-cells such that

∞

X

j=1

c(K_i) < ε.

A bounded set C whose boundary has measure 0 is called Jordan-measurable.

• If A = A1∪ A2∪ A3∪ · · · and each Ai has measure 0, then A has measure 0.

• If A is compact and has measure 0, then A has content 0.

7. • If Ψ is another partition of unity, subordinate to an admissible cover F of A, then X

ψ∈Ψ

Z

A

ψ · |f | also converges, and

X

ϕ∈Φ

Z

A

ϕ · f =X

ψ∈Ψ

Z

A

ψ · f.

• If A and f are bounded, then f is integrable in the extended sense.

• If A is Jordan-measurable and f is bounded, then this definition of Z

A

f agrees with the old one.

8. (Sard’s Theorem) If U ⊂ Rⁿ is open, g : U → Rⁿ ∈ C¹(U ), and if B = {x ∈ U | det Dg(x) = 0}, then g(B) has measure 0.

Tensors on vector space

1. Familiar with k-tensor on a vector space V, tensor product of two tensors, the dual space of vector space V, the inner product on a vector space V, the basis ofT ^k(V ), the alternating k-tensor on V and the space Λ^k(V ) of alternating k-tensors on V.

2. Know that

• if T ∈T ^k(V ), then Alt (T ) defined by Alt (T )(v1, . . . , vk) = 1

k!

X

σ∈Sk

sgn σ · T (vσ(1), . . . , vσ(k)) ∈ Λ^k(V ),

where S_k is the set of all permutations of numbers 1 to k.

• If w ∈ Λ^k(V ), then Alt (w) ∈ Λ^k(V ).

• If T ∈T^k(V ), then Alt (Alt (T )) = Alt (T ).

3. If w ∈ Λ^k(V ) and η ∈ Λ^{`</sup>(V ), we define the wedge product w ∧ η ∈ Λ<sup>k+`}(V ) by w ∧ η = (k + `)!

k! `! Alt (w ⊗ η).

Note that ∧ satisfies the following properties.

• w ∧ η = (−1)^k`η ∧ w,

• f^∗(w ∧ η) = f^∗(w) ∧ f^∗(η).

• If w ∈ Λ^k(V ), η ∈ Λ<sup>`</sup>(V ), and θ ∈ Λ<sup>m</sup>(V ), then (w ∧ η) ∧ θ = w ∧ (η ∧ θ) = (k + ` + m)!

k! `! m! Alt (w ⊗ η ⊗ θ) = w ∧ η ∧ θ.

4. If {v_i}ⁿ_i=1 is a basis for V and {ϕ_i}ⁿ_i=1 is the dual basis such that

ϕ_i(v_j) = δ_ij =

(1 if i = j, 0 if i 6= j.

Then the set of all

{ϕ_i1 ∧ · · · ∧ ϕ_ik | 1 ≤ i₁ < i₂ < . . . < i_k ≤ n}

is a basis of Λ^k(V ), which therefore has dimension

n k



= n!

k! (n − k)!.

5. Let {v_i}ⁿ_i=1 be a basis for V, and let w ∈ Λⁿ(V ). If {w_i}ⁿ_i=1 are n vectors in V with w_i =

n

X

j=1

a_ijv_j, then

w(w₁, . . . , w_n) = det(a_ij) · w(v₁, . . . , v_n).

Fields and forms

1. Familiar with definitions of a vector field on Rⁿ, a k-form on Rⁿ, the tangent space Rⁿp of Rⁿ at p, the (push-forward) linear transformation f∗ : [

p∈Rⁿ

Rⁿp → [

p∈Rⁿ

R^mf (p) defined by f∗(v_p) = Df (p)(v)

f (p) for v_p = (p, v) ∈ Rⁿp and p ∈ Rⁿ, and the (pullback) linear transformation f^∗ : Λ^k(R^mf (p)) → Λ^k(Rⁿp) defined by f^∗w(p)(v₁, . . . , v_k) = w(f (p))(f_∗(v₁), . . . , f_∗(v_k)) for v₁, . . . , v_k ∈ Rⁿp.

2. Let x = (x¹, . . . , xⁿ) ∈ Rⁿ and y = (y¹, . . . , y^m) ∈ R^m. If f : Rⁿ→ R^m is differentiable then

• f^∗(dy^`) =

n

X

i=1

D_if^`dxⁱ =

n

X

i=1

∂y^`

∂xⁱ dxⁱ.

• f^∗(dy^{`1</sup> ∧ dy<sup>`2}) =X

i<j

∂(y^{`1</sup>, y<sup>`2})

∂(xⁱ, x^j) dxⁱ∧ dx^j.

• f^∗(dy^{`1</sup> ∧ · · · ∧ dy<sup>`k}) = X

i1<···<ik

∂(y^{`1</sup>, . . . , y<sup>`k})

∂(xⁱ¹, . . . , x^ik)dxⁱ¹ ∧ · · · ∧ dx^ik

• f^∗(g · w) = (g ◦ f ) · f^∗(w), where g : R^m → R^p is a function (i.e. 0-form) on R^m.

• f^∗(w ∧ η) = f^∗w ∧ f^∗η.

3. If f : Rⁿ→ Rⁿ is differentiable, then

f^∗(hdx¹∧ · · · ∧ dxⁿ) = (h ◦ f ) (det Df ) dx¹∧ · · · ∧ dxⁿ.

4. For each k ≥ 0 and for each

w = X

1≤i1<i2<···<ik≤n

w_i1,...,ikdxⁱ¹ ∧ · · · ∧ dx^ik ∈ Λ^k(Rⁿ),

define the linear transformation d : Λ^k(Rⁿ) → Λ^k+1(Rⁿ) by

dw = X

1≤i1<i2<···<ik≤n n

X

α=1

D_α(w_i1,...,ik) dx^α∧ dxⁱ¹ ∧ · · · ∧ dx^ik.

Let w be a k-form and η be an `-form on R^m. Then

• d(w ∧ η) = dw ∧ η + (−1)^kw ∧ dη.

• d(dw) = 0, i.e. d(Λ^k(R^m)) is a subset of the kernel of d : Λ^k+1(R^m) → Λ^k+2(R^m).

Briefly, d² = 0.

• If f : Rⁿ → R^m is differentiable, then f^∗(dw) = d(f^∗w).

• w is called closed if dw = 0 and is called exact if w = dτ, for some (k − 1) form τ.

5. (Poincar´e Lemma) Let A ⊂ Rⁿ be an open set star-shaped with respect to 0, i.e. for each x ∈ A, the line segment from 0 to x is contained in A. Then every closed form on A is exact.

6. (Stokes’ Theorem) If w is a (k − 1)-form on an open set A ⊂ Rⁿ and c is a k-chain in A, then

Z

c

dw = Z

∂c

w.