Tangent space is all at a certain pointTangent vectorConstituentlinear space。Tangent space isDifferential manifoldThe vector space connected at a point,euclidean space The extension of tangent line of smooth curve and tangent plane of smooth surface.The tangent vector istangentOfDirection vector。For the curve Γ: x=φ (t), t ∈ [a, b], x ∈ Rn, when the vector φ ′ (tzero）When exists and is not equal to zero, it is called Γ at the point φ (tzero）Tangent vector at. φ '(tzero)/|φ′(tzero）|Is called curve Γ in φ (tzero）The unit tangent vector at.Through φ (tzero）And the tangent vector φ ′ (tzero）The parallel straight line is Γ at φ (tzero）Tangent at.
Linear space is also called vector space.It is the central content and one of the basic concepts of linear algebra.Let V be a non empty set and P be a field.If:
1. An operation is defined in V, which is called addition, that is, any two elements α and β in V correspond to an element α+β uniquely determined in V according to a certain rule, which is called the sum of α and β.
2. An operation is defined between the elements of P and V, which is called scalar multiplication (also called quantitative multiplication). That is, for any element α in V and any element k in P, a certain rule corresponds to a uniquely determined element k α in V, which is called the product of k and α.[1]
3. Addition and scalar multiplication meet the following conditions:
1) α+β=β+α, for any α, β ∈ V
2) α+(β+γ)=(α+β)+γ, for any α, β, γ ∈ V
3) There is an element 0 ∈ V, α+0=α for all α ∈ V, and element 0 is called the zero element of V
4) For any α ∈ V, there exists β ∈ V so that α+β=0, β is called the negative element of α, which is denoted as - α
5) For the unit element 1 in P, there is 1 α=α (α ∈ V)
6) For any k, l ∈ P, α ∈ V, there is (kl) α=k (l α)
7) For any k, l ∈ P, α ∈ V, there is (k+l) α=k α+l α
8) For any k ∈ P, α, β ∈ V, k (α+β)=k α+k β,
Then V is called a linear space, or vector space, on the field P.The element in V is called vector, the zero element of V is called zero vector, and P is called the base field of linear space. When P is a real number field, V is called a real linear space. When P is a complex number field, V is called a complex linear space.For example, if V is a set of all vectors (directed line segments) in 3D geometric space, and P is the real number field R, then V is about vector addition (that isParallelogram rule）The multiplication of sum and vector forms a linear space on the real number field R.For another example, if V is the set Mmn (P) composed of all m × n matrices over the number field P, and the addition and scalar multiplication of V are respectively the addition of matrix and the multiplication of number and matrix, then Mmn (P) is a linear space over the number field PThe vector in V is the m × n matrix.For another example, the set P formed by all n-ary vectors (a1, a2,..., an) on the field P for addition: (a1, a2,..., an)+(b1, b2,..., bn)=(a1+b1, a2+b2,..., an+bn) and scalar multiplication: λ (a1, a2,..., an)=(λ a1, λ a2,..., λ an) constitutes a linear space on the field P, which is called n-ary vector space on the field P.
Linear space is a mathematical concept abstracted after investigating the essential attributes of a large number of mathematical objects (such as vectors in geometry and physics, n-ary vectors, matrices, polynomials in algebra, functions in analysis, etc.). Many research objects in modern mathematics, such asNormed linear space, modules and so on are closely related to linear space.Its theories and methods have penetrated into many fields of natural science and engineering technology.Hamilton(Hamilton, W.R.) first introduced the word vector, and pioneered vector theory and vector computing.Glassman(Grassmann, H.G.) was the first to put forward the system theory of multidimensional Euclidean space.From 1844 to 1847, he andCauchy(Cauchy, A. - L.) respectively proposed an abstract n-dimensional space that is independent of all spaces and becomes a pure mathematical concept.Tplitz(Toeplitz, O.) extended the main theorems of linear algebra to general linear spaces on arbitrary fields.[2]
Tangent vector
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There are many definitions of tangent vector.Intuitively, if themanifoldIs a 3D spacecurved surface, the tangent vector at each point is the vector tangent to the surface, and the tangent space is the plane tangent to the surface.In general, because all manifolds can be embeddedEuclidean spaceA tangent space can also be understood as an affine subspace of the Euclidean space where point is tangent to manifold.The better definition of tangent space does not depend on this embedding. For example, the tangent vector can be defined as the equivalent class of the curve passing through the point, or it can be defined as the equivalent class of the curve passing through the pointSmooth functionDerivation in a certain direction at the point.But all these definitions are equivalent.
Set M YesDifferentiableOfmanifold, p is the last point of M, allTangent vectorAll Zhang Chenglinear spaceIs called the tangent space of M at p, and is recorded as Tp(M). If p is a smooth point, then Tp(M) OfdimensionIs the dimension of manifold M.[3]
Detailed definition of tangent space
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Tangent space isDifferential manifoldThe vector space connected at a point,euclidean space The extension of tangent line of smooth curve and tangent plane of smooth surface.If M isN-dimensional differential manifold, p ∈ M, mark C∞(p) Is C defined in a neighborhood of p∞The set of differentiable functions, then the function X suitable for the following conditionsp：C∞(p) → R is called the tangent vector of M at p:[4]
1. For f, g ∈ C∞(p) , if there is a neighborhood U of p in M, so that f | U=g | U, then Xp(f)=Xp(g).
2. For f, g ∈ C∞(p) , α, β ∈ R, have:
At this time, C∞(p) The operations of functions in are defined as follows:
(α f+β g) (q)=α f (q)+β g (q) ∈ R, when f (q), g (q) are defined.
3. For f, g ∈ C (p), there are: Xp(f×g)=f(p)Xp(g)+g(p)Xp(f)，
Where f × g is the multiplication of ordinary functions, that is, (f × g) (q)=f (q) g (q).
The set of all tangent vectors of differential manifold M at p ∈ M is marked as TpM，For Xp，Yp∈TpM. α ∈ R and f ∈ C∞(p) , set:
So TpM is the n-dimensional vector space on the real number field R, which is called the tangent space of the differential manifold M at p.
Tangent space TpRepresentation of tangent vector in M: Let (U, φ) be the card containing point p in M, and the local coordinates on U are:
For i=1, 2,..., n, if:
Where (uone，utwo，…，un）Yes RnMiddle coordinate, then:
