Tangent Space
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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the tangent space of a
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''
tangent line In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...
s'' to curves in two dimensions. In the context of physics the tangent space to a manifold at a point can be viewed as the space of possible velocities for a particle moving on the manifold.


Informal description

In
differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
, one can attach to every point x of a
differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
a ''tangent space''—a real
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
that intuitively contains the possible directions in which one can tangentially pass through x . The elements of the tangent space at x are called the ''
tangent vector In mathematics, a tangent vector is a vector that is tangent to a curve or surface at a given point. Tangent vectors are described in the differential geometry of curves in the context of curves in R''n''. More generally, tangent vectors are eleme ...
s'' at x . This is a generalization of the notion of a
vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...
, based at a given initial point, in a
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
. The
dimension In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
of the tangent space at every point of a
connected Connected may refer to: Film and television * ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular'' * '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film * ''Connected'' (2015 TV ...
manifold is the same as that of the
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
itself. For example, if the given manifold is a 2 -
sphere A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
, then one can picture the tangent space at a point as the plane that touches the sphere at that point and is
perpendicular In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It can ...
to the sphere's radius through the point. More generally, if a given manifold is thought of as an embedded
submanifold In mathematics, a submanifold of a manifold ''M'' is a subset ''S'' which itself has the structure of a manifold, and for which the inclusion map satisfies certain properties. There are different types of submanifolds depending on exactly which ...
of
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
, then one can picture a tangent space in this literal fashion. This was the traditional approach toward defining
parallel transport In geometry, parallel transport (or parallel translation) is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection (a covariant derivative or connection (vector bundle), c ...
. Many authors in
differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
and
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
use it. More strictly, this defines an affine tangent space, which is distinct from the space of tangent vectors described by modern terminology. In
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, in contrast, there is an intrinsic definition of the ''tangent space at a point'' of an
algebraic variety Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Mo ...
V that gives a vector space with dimension at least that of V itself. The points p at which the dimension of the tangent space is exactly that of V are called ''non-singular'' points; the others are called ''singular'' points. For example, a curve that crosses itself does not have a unique tangent line at that point. The singular points of V are those where the "test to be a manifold" fails. See
Zariski tangent space In algebraic geometry, the Zariski tangent space is a construction that defines a tangent space at a point ''P'' on an algebraic variety ''V'' (and more generally). It does not use differential calculus, being based directly on abstract algebra, an ...
. Once the tangent spaces of a manifold have been introduced, one can define vector fields, which are abstractions of the velocity field of particles moving in space. A vector field attaches to every point of the manifold a vector from the tangent space at that point, in a smooth manner. Such a vector field serves to define a generalized
ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast w ...
on a manifold: A solution to such a differential equation is a differentiable
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (ge ...
on the manifold whose derivative at any point is equal to the tangent vector attached to that point by the vector field. All the tangent spaces of a manifold may be "glued together" to form a new differentiable manifold with twice the dimension of the original manifold, called the ''
tangent bundle In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of ...
'' of the manifold.


Formal definitions

The informal description above relies on a manifold's ability to be embedded into an ambient vector space \mathbb^ so that the tangent vectors can "stick out" of the manifold into the ambient space. However, it is more convenient to define the notion of a tangent space based solely on the manifold itself. There are various equivalent ways of defining the tangent spaces of a manifold. While the definition via the velocity of curves is intuitively the simplest, it is also the most cumbersome to work with. More elegant and abstract approaches are described below.


Definition via tangent curves

In the embedded-manifold picture, a tangent vector at a point x is thought of as the ''velocity'' of a
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (ge ...
passing through the point x . We can therefore define a tangent vector as an equivalence class of curves passing through x while being tangent to each other at x . Suppose that M is a C^
differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
(with smoothness k \geq 1 ) and that x \in M . Pick a
coordinate chart In topology, a branch of mathematics, a topological manifold is a topological space that locally resembles real ''n''-dimensional Euclidean space. Topological manifolds are an important class of topological spaces, with applications throughout mathe ...
\varphi: U \to \mathbb^ , where U is an open subset of M containing x . Suppose further that two curves \gamma_,\gamma_: (- 1,1) \to M with (0) = x = (0) are given such that both \varphi \circ \gamma_,\varphi \circ \gamma_: (- 1,1) \to \mathbb^ are differentiable in the ordinary sense (we call these ''differentiable curves initialized at x ''). Then \gamma_ and \gamma_ are said to be ''equivalent'' at 0 if and only if the derivatives of \varphi \circ \gamma_ and \varphi \circ \gamma_ at 0 coincide. This defines an
equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relation ...
on the set of all differentiable curves initialized at x , and
equivalence class In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...
es of such curves are known as ''tangent vectors'' of M at x . The equivalence class of any such curve \gamma is denoted by \gamma'(0) . The ''tangent space'' of M at x , denoted by T_ M , is then defined as the set of all tangent vectors at x ; it does not depend on the choice of coordinate chart \varphi: U \to \mathbb^ . To define vector-space operations on T_ M , we use a chart \varphi: U \to \mathbb^ and define a
map A map is a symbolic depiction emphasizing relationships between elements of some space, such as objects, regions, or themes. Many maps are static, fixed to paper or some other durable medium, while others are dynamic or interactive. Although ...
\mathrm_: T_ M \to \mathbb^ by (\gamma'(0)) := \left. \frac \varphi \circ \gamma)(t)\_, where \gamma \in \gamma'(0) . The map \mathrm_ turns out to be
bijective In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
and may be used to transfer the vector-space operations on \mathbb^ over to T_ M , thus turning the latter set into an n -dimensional real vector space. Again, one needs to check that this construction does not depend on the particular chart \varphi: U \to \mathbb^ and the curve \gamma being used, and in fact it does not.


Definition via derivations

Suppose now that M is a C^ manifold. A real-valued function f: M \to \mathbb is said to belong to (M) if and only if for every coordinate chart \varphi: U \to \mathbb^ , the map f \circ \varphi^: \varphi \subseteq \mathbb^ \to \mathbb is infinitely differentiable. Note that (M) is a real
associative algebra In mathematics, an associative algebra ''A'' is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field ''K''. The addition and multiplic ...
with respect to the
pointwise product In mathematics, the pointwise product of two functions is another function, obtained by multiplying the images of the two functions at each value in the domain. If and are both functions with domain and codomain , and elements of can be mul ...
and sum of functions and scalar multiplication. A ''
derivation Derivation may refer to: Language * Morphological derivation, a word-formation process * Parse tree or concrete syntax tree, representing a string's syntax in formal grammars Law * Derivative work, in copyright law * Derivation proceeding, a proc ...
'' at x \in M is defined as a
linear map In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \to W between two vect ...
D: (M) \to \mathbb that satisfies the Leibniz identity \forall f,g \in (M): \qquad D(f g) = D(f) \cdot g(x) + f(x) \cdot D(g), which is modeled on the
product rule In calculus, the product rule (or Leibniz rule or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions. For two functions, it may be stated in Lagrange's notation as (u \cdot v)' = u ' \cdot v + ...
of calculus. (For every identically constant function f=\text, it follows that D(f)=0 ). Denote T_ M the set of all derivations at x. Setting * (D_1+D_2)(f) := _1(f) + _2(f) and * (\lambda \cdot D)(f) := \lambda \cdot D(f) turns T_ M into a vector space.


Generalizations

Generalizations of this definition are possible, for instance, to
complex manifold In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic. The term complex manifold is variously used to mean a com ...
s and
algebraic varieties Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Mo ...
. However, instead of examining derivations D from the full algebra of functions, one must instead work at the level of germs of functions. The reason for this is that the structure sheaf may not be
fine Fine may refer to: Characters * Sylvia Fine (''The Nanny''), Fran's mother on ''The Nanny'' * Officer Fine, a character in ''Tales from the Crypt'', played by Vincent Spano Legal terms * Fine (penalty), money to be paid as punishment for an offe ...
for such structures. For example, let X be an algebraic variety with structure sheaf \mathcal_ . Then the
Zariski tangent space In algebraic geometry, the Zariski tangent space is a construction that defines a tangent space at a point ''P'' on an algebraic variety ''V'' (and more generally). It does not use differential calculus, being based directly on abstract algebra, an ...
at a point p \in X is the collection of all \mathbb -derivations D: \mathcal_ \to \mathbb , where \mathbb is the
ground field In mathematics, a ground field is a field ''K'' fixed at the beginning of the discussion. Use It is used in various areas of algebra: In linear algebra In linear algebra, the concept of a vector space may be developed over any field. In algeb ...
and \mathcal_ is the stalk of \mathcal_ at p .


Equivalence of the definitions

For x \in M and a differentiable curve \gamma: (- 1,1) \to M such that \gamma (0) = x, define (f) := (f \circ \gamma)'(0) (where the derivative is taken in the ordinary sense because f \circ \gamma is a function from (- 1,1) to \mathbb ). One can ascertain that D_(f) is a derivation at the point x, and that equivalent curves yield the same derivation. Thus, for an equivalence class \gamma'(0), we can define (f) := (f \circ \gamma)'(0), where the curve \gamma \in \gamma'(0) has been chosen arbitrarily. The map \gamma'(0) \mapsto D_ is a vector space isomorphism between the space of the equivalence classes \gamma'(0) and that of the derivations at the point x.


Definition via cotangent spaces

Again, we start with a C^\infty manifold M and a point x \in M . Consider the
ideal Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considere ...
I of C^\infty(M) that consists of all smooth functions f vanishing at x , i.e., f(x) = 0 . Then I and I^2 are both real vector spaces, and the quotient space I / I^2 can be shown to be isomorphic to the
cotangent space In differential geometry, the cotangent space is a vector space associated with a point x on a smooth (or differentiable) manifold \mathcal M; one can define a cotangent space for every point on a smooth manifold. Typically, the cotangent space, T ...
T^_x M through the use of
Taylor's theorem In calculus, Taylor's theorem gives an approximation of a ''k''-times differentiable function around a given point by a polynomial of degree ''k'', called the ''k''th-order Taylor polynomial. For a smooth function, the Taylor polynomial is the t ...
. The tangent space T_x M may then be defined as the
dual space In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by const ...
of I / I^2 . While this definition is the most abstract, it is also the one that is most easily transferable to other settings, for instance, to the
varieties Variety may refer to: Arts and entertainment Entertainment formats * Variety (radio) * Variety show, in theater and television Films * ''Variety'' (1925 film), a German silent film directed by Ewald Andre Dupont * ''Variety'' (1935 film), ...
considered in
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
. If D is a derivation at x , then D(f) = 0 for every f \in I^2 , which means that D gives rise to a linear map I / I^2 \to \mathbb . Conversely, if r: I / I^2 \to \mathbb is a linear map, then D(f) := r\left((f - f(x)) + I^2\right) defines a derivation at x . This yields an equivalence between tangent spaces defined via derivations and tangent spaces defined via cotangent spaces.


Properties

If M is an open subset of \mathbb^ , then M is a C^ manifold in a natural manner (take coordinate charts to be identity maps on open subsets of \mathbb^ ), and the tangent spaces are all naturally identified with \mathbb^ .


Tangent vectors as directional derivatives

Another way to think about tangent vectors is as
directional derivative In mathematics, the directional derivative of a multivariable differentiable (scalar) function along a given vector v at a given point x intuitively represents the instantaneous rate of change of the function, moving through x with a velocity s ...
s. Given a vector v in \mathbb^ , one defines the corresponding directional derivative at a point x \in \mathbb^ by : \forall f \in (\mathbb^): \qquad (D_ f)(x) := \left. \frac
(x + t v) X, or x, is the twenty-fourth and third-to-last letter in the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''"ex"'' (pronounced ), ...
\_ = \sum_^ v^ (x). This map is naturally a derivation at x . Furthermore, every derivation at a point in \mathbb^ is of this form. Hence, there is a one-to-one correspondence between vectors (thought of as tangent vectors at a point) and derivations at a point. As tangent vectors to a general manifold at a point can be defined as derivations at that point, it is natural to think of them as directional derivatives. Specifically, if v is a tangent vector to M at a point x (thought of as a derivation), then define the directional derivative D_ in the direction v by : \forall f \in (M): \qquad (f) := v(f). If we think of v as the initial velocity of a differentiable curve \gamma initialized at x , i.e., v = \gamma'(0) , then instead, define D_ by : \forall f \in (M): \qquad (f) := (f \circ \gamma)'(0).


Basis of the tangent space at a point

For a C^ manifold M , if a chart \varphi = (x^,\ldots,x^): U \to \mathbb^ is given with p \in U , then one can define an ordered basis \left\ of T_ M by : \forall i \in \, ~ \forall f \in (M): \qquad (f) := \left( \frac \Big( f \circ \varphi^ \Big) \right) \Big( \varphi(p) \Big) . Then for every tangent vector v \in T_ M , one has : v = \sum_^ v^ \left. \frac \_. This formula therefore expresses v as a linear combination of the basis tangent vectors \left. \frac \_ \in T_ M defined by the coordinate chart \varphi: U \to \mathbb^ .


The derivative of a map

Every smooth (or differentiable) map \varphi: M \to N between smooth (or differentiable) manifolds induces natural
linear map In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \to W between two vect ...
s between their corresponding tangent spaces: : \mathrm_: T_ M \to T_ N. If the tangent space is defined via differentiable curves, then this map is defined by : (\gamma'(0)) := (\varphi \circ \gamma)'(0). If, instead, the tangent space is defined via derivations, then this map is defined by : mathrm_(D)f) := D(f \circ \varphi). The linear map \mathrm_ is called variously the ''derivative'', ''total derivative'', ''differential'', or ''pushforward'' of \varphi at x . It is frequently expressed using a variety of other notations: : D \varphi_, \qquad (\varphi_)_, \qquad \varphi'(x). In a sense, the derivative is the best linear approximation to \varphi near x . Note that when N = \mathbb , then the map \mathrm_: T_ M \to \mathbb coincides with the usual notion of the differential of the function \varphi . In
local coordinates Local coordinates are the ones used in a ''local coordinate system'' or a ''local coordinate space''. Simple examples: * Houses. In order to work in a house construction, the measurements are referred to a control arbitrary point that will allow ...
the derivative of \varphi is given by the Jacobian. An important result regarding the derivative map is the following: This is a generalization of the
inverse function theorem In mathematics, specifically differential calculus, the inverse function theorem gives a sufficient condition for a function to be invertible in a neighborhood of a point in its domain: namely, that its ''derivative is continuous and non-zero at ...
to maps between manifolds.


See also

* Coordinate-induced basis *
Cotangent space In differential geometry, the cotangent space is a vector space associated with a point x on a smooth (or differentiable) manifold \mathcal M; one can define a cotangent space for every point on a smooth manifold. Typically, the cotangent space, T ...
*
Differential geometry of curves Differential geometry of curves is the branch of geometry that deals with smooth curves in the plane and the Euclidean space by methods of differential and integral calculus. Many specific curves have been thoroughly investigated using the sy ...
* Exponential map *
Vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...


Notes


References

* . * . * .


External links


Tangent Planes
at MathWorld {{DEFAULTSORT:Tangent Space Differential topology Differential geometry