Let X and Y be sets. A bijection from X to Y is a function f:X→Y with the following property: for every element y∈Y there exists exactly one element x∈X such that f(x)=y.
Let r∈N, and let σ∈Sr, where Sr is the set from the permutation definition. An inversion of σ is a pair (i,j) such that 1≤i<j≤r and σ(i)>σ(j). Let N(σ) denote the number of inversions of σ. The sign of σ…
Let n∈N. We say that n is even if there exists a natural number q∈N such that n=2q. We say that n is odd if there exists a natural number q∈N such that n=2q−1.
Let m,n∈N. Let A=(Aαi) be an m×n matrix with real entries, and let v=(v1,…,vn)∈Rn. The product vector Av∈Rm is defined by (Av)α=∑i=1nAαivi for every α∈{1,…,m}.
Let m,n,p∈N. Let A=(Aαi) be an m×n matrix with real entries, and let B=(Biβ) be an n×p matrix with real entries. The product matrix AB is the m×p matrix whose (α,β) entry is defined by…
Let n,m,p∈N. Let U⊆Rn, V⊆Rm, and W⊆Rp be open subsets. Let f:U→V and g:V→W be C1 maps. Then the composition g∘f:U→W is again of class C1. Moreover, for every a∈U, the Jacob…
Let n,m,k∈N. Let U⊆Rn and V⊆Rm be open subsets, let F:U→V be a C1 map, and let ω be a differential k-form on V. The pullback of ω by F is the differential k-form F∗ω on U defined by…
Let n∈N and let k,ℓ∈N∪{0}. Let U⊆Rn be open. Let α be a differential k-form on U, and let β be a differential ℓ-form on U. The wedge product α∧β is the differential (k+ℓ)-form on…
Let n∈N, let U⊆Rn be open, and let k∈N∪{0}. A differential k-form on U is an assignment ω which to each point x∈U assigns an alternating k-linear form ωx on Rn. For vectors…
Let n,k∈N. A function ω:(Rn)k→R is called a k-linear form on Rn if for each index r∈{1,…,k}, for every choice of vectors v1,…,vr−1,u,w,vr+1,…,vk∈Rn, and for every scalars…
Let n,m∈N. Let U⊆Rn be open, and let f=(f1,…,fm):U→Rm. We say that f is of class C1 on U if each coordinate function fj:U→R is continuous at every point of U, and if for every j∈{1,…,m} and eve…
Let n,m∈N. Let E⊆Rn, let f:E→Rm, let a∈E, and write f=(f1,…,fm). We say that f is continuous at a if for every ε>0 there exists δ>0 such that for every point x=(x1,…,xn)∈E, if…
Let n∈N and let U⊆Rn. We say that U is open in Rn if for every point x=(x1,…,xn)∈U there exists a real number r>0 such that every point y=(y1,…,yn)∈Rn satisfying ∑i=1n(yi−xi)2<r2 al…
Let (xn)n=1∞ be a sequence of real numbers. A subsequence of (xn) is a sequence of the form (xnk)k=1∞, where (nk)k=1∞ is a strictly increasing sequence of positive integers.