
Contraction Mapping Theorem
Let X be a complete metric space. If ƒ : X → X is a contraction, then ƒ has a unique fixed point.

CauchySchwartz for matrices
Let A ∈ R^{n×m} and b ∈ R^{m×k}. Then AB ∈ R^{n×k} and ‖AB‖ ≤ ‖A‖ ‖B‖

Chain Rule
Let U open in R^{n}, V open in R^{m} with F : U → V and G : V → R^{k}. If F is differentiable at u ∈ U and F is differentiable at F(u) = v ∈ V, then G ∘ F is differentiable at u. Furthermore if T : R^{n} → R^{m} is the first order affine approximation to F at u and S : R^{m} → R^{k} is the first order affine approximation to G at v then S ∘ T is the affine function which approximates G ∘ F to first order at u.

Mean Value Theorem for Real Valued Functions
Let U be an open subset of R^{n} and ƒ : U → R differentiable. If the line segment J from a to b is contained in U, then there exists c ∈ J with ƒ(b) − ƒ(a) = ∇ƒ(c) • (b − a).

Open Mapping Theorem
Let U be open in R^{n} and F ∈ C^{1}(U, R^{n}). If F is smooth, then F is an open function.

Inverse Function Theorem
Let F ∈ C^{1}(W, R^{n}) where W is open in R^{n}. If p ∈ W is a smooth point of F, then there exists open U ∋ p with U ⊆ W and an open set V such that F : U → V is a smooth homeomorphism. Also F^{1} ∈ C^{1}(V, U) and D[F^{1}](F(a)) = [DF(a)]^{1} for any a ∈ U.

Implicit Function Theorem
 For a function F ∈ C^{1}(Ω, R^{m}) (Ω open in R^{n+m}) and a ∈ R^{n} and b ∈ R^{m} with F(a, b) = 0. If D_{2}F(a, b) has rank m, then there exists open U ∋ a in R^{n} and open V ∋ (a, b) in R^{n+m} and a function F ∈ C^{1}(U, R^{m}) such that
 {(x, y) ∈ V : F(x, y) = 0} = {(x, G(x)) : x ∈ U}.
 In addition, DG(x) = [D_{2}F(x, G(x))]^{1}D_{1}F(x, G(x))

Lagrange Remainder Theorem
 Let ƒ ∈ C^{m+1}(U, R) where U is open in R^{n}. If the line segment bx ⊂ U, then
 ƒ(x) = p_{m}(x) + 1/(m + 1)! D^{m+1}ƒ(c)(u)^{m+1}
 where u = x − b and c ∈ bx.

Lagrange Multipliers Theorem
 Let F ∈ C^{1}(Ω, R^{m}) where Ω is open in R^{n+m} and S = {x ∈ Ω : F(x) = 0}. For a differentiable function φ : Ω → R, if s ∈ S is a local extreme of φ_{S}, the either DF(s) has rank less than m, or there exists a nonzero vector v ∈ R^{m} (written horizontally) such that
 ∇φ(s) = vDF(s)

Don’t Sweat the Small Stuff Lemma
Let I be a closed interval in R^{n}, A ⊂ I with volA = 0, and ƒ : I → R is a bounded function. If ƒ has the property that for any closed interval J ⊆ I, ƒ_{J} is integrable whenever J ∩ A = ∅, then ƒ is integrable, ƒ_{I−A} is integrable, and ∫_{I} ƒ = ∫_{I−A} ƒ.

Fubini’s Theorem
Let N and Q be closed intervals of R^{n} and R^{m} respectively, and let I = N × Q. Let ƒ : I → R be integrable. If for each y ∈ Q, the function g_{y} : N → R defined by g_{y}(x) = ƒ(x, y) is integrable, then the function h : Q → R defined by h(y) = ∫_{N} g_{y} is integrable and ∫_{Q} h = ∫_{I} ƒ.

Inner Cube Lemma
 Consider a compact A ⊂ U, an open set in R^{n}, and F ∈ C^{1}(U, R^{n}), a smooth one to one function. For any ε > 0, there exists a δ > 0 such that if C is an ncube with diamC < δ, C ∩ A ≠ ∅ and a ∈ C, then C ⊂ U and
 abs[volF(C)/volC − J_{F}(a)] < ε
 where J_{F}(a) = detDF(a) is the Jacobian.

Change of Variables
 Let U be an open set in R^{n} and Φ ∈ C^{1}(U, R^{n}), a smooth one to one function. If D ⊂ U is a compact Jordan domain and ƒ : φ(D) → R is continuous, then
 ∫_{Φ(D)} ƒ = ∫_{D} ƒ ∘ Φ J_{Φ}
 where J_{Φ} = detDΦ is the Jacobian.

