In

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 vectors'' at $x$. This is a generalization of the notion of a bound vector in a mathematics
Mathematics (from Greek: ) includes the study of such topics as quantity (number theory), structure (algebra), space (geometry), and change (analysis). It has no generally accepted definition.
Mathematicians seek and use patterns to formulate ...

, the tangent space of a manifold
The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surface.
In mathematics, a manifold is a topological space that locally resembles Euclidean space ...

facilitates the generalization of vectors from affine space
In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related ...

s to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector that gives the displacement
Displacement may refer to:
Physical sciences
Mathematics and Physics
*Displacement (geometry), is the difference between the final and initial position of a point trajectory (for instance, the center of mass of a moving object). The actual path c ...

of the one point from the other.
Informal description

. A vector in this tangent space represents a possible velocity at $x$. After moving in that direction to a nearby point, the velocity would then be given by a vector in the tangent space of that point—a different tangent space that is not shown. Indifferential geometry
Differential geometry is a mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The theory of plane and space curves and surfaces in th ...

, 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 linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an at ...

a ''tangent space''—a real vector space#REDIRECT Vector space#REDIRECT Vector space
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...Euclidean space
Euclidean space is the fundamental space of classical geometry. Originally it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension, including the three-di ...

. The dimension
thumb
, 236px
, The first four spatial dimensions, represented in a two-dimensional picture.
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to s ...

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 ser ...

manifold is the same as that of the manifold
The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surface.
In mathematics, a manifold is a topological space that locally resembles Euclidean space ...

itself.
For example, if the given manifold is a $2$-sphere
of a sphere
A sphere (from Greek language, Greek —, "globe, ball") is a geometrical object in three-dimensional space that is the surface of a ball (viz., analogous to the circular objects in two dimensions, where a "circle" circumscribes its " ...

, 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, the property of being perpendicular (perpendicularity) is the relationship between two lines which meet at a right angle (90 degrees). The property extends to other related geometric objects.
A line is said to be perpend ...

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 pro ...

of Euclidean space
Euclidean space is the fundamental space of classical geometry. Originally it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension, including the three-di ...

, then one can picture a tangent space in this literal fashion. This was the traditional approach toward defining parallel transport
In geometry, parallel transport 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 on the tangent bundle), then this connection ...

. Many authors in differential geometry
Differential geometry is a mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The theory of plane and space curves and surfaces in th ...

and general relativity
General relativity, also known as the general theory of relativity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics. General relativity generalizes spec ...

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 pro ...

, 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. Mode ...

$V$ 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, and ...

.
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 on a manifold: A solution to such a differential equation is a differentiable curve 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'' of the manifold.
Formal definitions

The informal description above relies on a manifold's ability to be embedded into an ambient vector space $\backslash mathbb^$ 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#Topology, curve 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 linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an at ...

(with smoothness $k\; \backslash geq\; 1$) and that $x\; \backslash in\; M$. Pick a Manifold#Charts, atlases, and transition maps, coordinate chart $\backslash varphi:\; U\; \backslash to\; \backslash mathbb^$, where $U$ is an open set, open subset of $M$ containing $x$. Suppose further that two curves $\backslash gamma\_,\backslash gamma\_:\; (-\; 1,1)\; \backslash to\; M$ with $(0)\; =\; x\; =\; (0)$ are given such that both $\backslash varphi\; \backslash circ\; \backslash gamma\_,\backslash varphi\; \backslash circ\; \backslash gamma\_:\; (-\; 1,1)\; \backslash to\; \backslash mathbb^$ are differentiable in the ordinary sense (we call these ''differentiable curves initialized at $x$''). Then $\backslash gamma\_$ and $\backslash gamma\_$ are said to be ''equivalent'' at $0$ if and only if the derivatives of $\backslash varphi\; \backslash circ\; \backslash gamma\_$ and $\backslash varphi\; \backslash circ\; \backslash gamma\_$ at $0$ coincide. This defines an equivalence relation on the set of all differentiable curves initialized at $x$, and equivalence classes of such curves are known as ''tangent vectors'' of $M$ at $x$. The equivalence class of any such curve $\backslash gamma$ is denoted by $\backslash gamma\text{'}(0)$. 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 $\backslash varphi:\; U\; \backslash to\; \backslash mathbb^$.
Image:Tangentialvektor.svg, left, 200px, The tangent space $T\_\; M$ and a tangent vector $v\; \backslash in\; T\_\; M$, along a curve traveling through $x\; \backslash in\; M$.
To define vector-space operations on $T\_\; M$, we use a chart $\backslash varphi:\; U\; \backslash to\; \backslash mathbb^$ and define a Map (mathematics), map $\backslash mathrm\_:\; T\_\; M\; \backslash to\; \backslash mathbb^$ by $(\backslash gamma\text{'}(0))\; ~\; \backslash stackrel\; ~\; \backslash left.\; \backslash frac\; [(\backslash varphi\; \backslash circ\; \backslash gamma)(t)]\; \backslash \_,$ where $\backslash gamma\; \backslash in\; \backslash gamma\text{'}(0)$. Again, one needs to check that this construction does not depend on the particular chart $\backslash varphi:\; U\; \backslash to\; \backslash mathbb^$ and the curve $\backslash gamma$ being used, and in fact it does not.
The map $\backslash mathrm\_$ turns out to be bijective and may be used to transfer the vector-space operations on $\backslash mathbb^$ over to $T\_\; M$, thus turning the latter set into an $n$-dimensional real vector space.
Definition via derivations

Suppose now that $M$ is a $C^$ manifold. A real-valued function $f:\; M\; \backslash to\; \backslash mathbb$ is said to belong to $(M)$ if and only if for every coordinate chart $\backslash varphi:\; U\; \backslash to\; \backslash mathbb^$, the map $f\; \backslash circ\; \backslash varphi^:\; \backslash varphi[U]\; \backslash subseteq\; \backslash mathbb^\; \backslash to\; \backslash mathbb$ is infinitely differentiable. Note that $(M)$ is a real associative algebra with respect to the pointwise product and sum of functions and scalar multiplication. Pick a point $x\; \backslash in\; M$. A ''Derivation (abstract algebra), derivation'' at $x$ is defined as a linear map $D:\; (M)\; \backslash to\; \backslash mathbb$ that satisfies the Leibniz identity :$\backslash forall\; f,g\; \backslash in\; (M):\; \backslash qquad\; D(f\; g)\; =\; D(f)\; \backslash cdot\; g\; +\; f\; \backslash cdot\; D(g),$ which is modeled on the product rule of calculus. (For every identically constant function $f=\backslash text,$ it follows that $D(f)=0$). If we define addition and scalar multiplication on the set of derivations at $x$ by * $(D\_1+D\_2)(f)\; ~\; \backslash stackrel\; ~\; \_1(f)\; +\; \_2(f)$ and * $(\backslash lambda\; \backslash cdot\; D)(f)\; ~\; \backslash stackrel\; ~\; \backslash lambda\; \backslash cdot\; D(f)$, then we obtain a real vector space, which we define as the tangent space $T\_\; M$ of $M$ at $x$.Generalizations

Generalizations of this definition are possible, for instance, to complex manifolds and algebraic variety, algebraic varieties. However, instead of examining derivations $D$ from the full algebra of functions, one must instead work at the level of germ (mathematics), germs of functions. The reason for this is that the structure sheaf may not be injective sheaf#Fine sheaves, fine for such structures. For example, let $X$ be an algebraic variety with structure sheaf $\backslash mathcal\_$. Then theZariski 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, and ...

at a point $p\; \backslash in\; X$ is the collection of all $\backslash mathbb$-derivations $D:\; \backslash mathcal\_\; \backslash to\; \backslash mathbb$, where $\backslash mathbb$ is the ground field and $\backslash mathcal\_$ is the stalk (sheaf), stalk of $\backslash mathcal\_$ at $p$.
Equivalence of the definitions

For $x\; \backslash in\; M$ and a differentiable curve $\backslash gamma:\; (-\; 1,1)\; \backslash to\; M$ such that $\backslash gamma\; (0)\; =\; x,$ define $(f)\; ~\; \backslash stackrel\; ~\; (f\; \backslash circ\; \backslash gamma)\text{'}(0)$ (where the derivative is taken in the ordinary sense because $f\; \backslash circ\; \backslash gamma$ is a function from $(-\; 1,1)$ to $\backslash mathbb$). 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 $\backslash gamma\text{'}(0),$ we can define $(f)\; \backslash stackrel\; (f\; \backslash circ\; \backslash gamma)\text{'}(0),$ where the curve $\backslash gamma\; \backslash in\; \backslash gamma\text{'}(0)$ has been chosen arbitrarily. The map $\backslash gamma\text{'}(0)\; \backslash mapsto\; D\_$ is a vector space isomorphism between the space of the equivalence classes $\backslash gamma\text{'}(0)$ and that of the derivations at the point $x.$Definition via cotangent spaces

Again, we start with a $C^\backslash infty$ manifold $M$ and a point $x\; \backslash in\; M$. Consider the ideal (ring theory), ideal $I$ of $C^\backslash infty(M)$ that consists of all smooth functions $f$ vanishing at $x$, i.e., $f(x)\; =\; 0$. Then $I$ and $I^2$ are real vector spaces, and $T\_x\; M$ may be defined as the dual space of the quotient space (linear algebra), quotient space $I\; /\; I^2$. This latter quotient space is also known as the ''cotangent space'' of $M$ at $x$. While this definition is the most abstract, it is also the one that is most easily transferable to other settings, for instance, to the algebraic variety, varieties considered inalgebraic 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 pro ...

.
If $D$ is a derivation at $x$, then $D(f)\; =\; 0$ for every $f\; \backslash in\; I^2$, which means that $D$ gives rise to a linear map $I\; /\; I^2\; \backslash to\; \backslash mathbb$. Conversely, if $r:\; I\; /\; I^2\; \backslash to\; \backslash mathbb$ is a linear map, then $D(f)\; ~\; \backslash stackrel\; ~\; r\backslash left((f\; -\; f(x))\; +\; I^2\backslash right)$ 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 $\backslash mathbb^$, then $M$ is a $C^$ manifold in a natural manner (take coordinate charts to be Identity function, identity maps on open subsets of $\backslash mathbb^$), and the tangent spaces are all naturally identified with $\backslash mathbb^$.Tangent vectors as directional derivatives

Another way to think about tangent vectors is as directional derivatives. Given a vector $v$ in $\backslash mathbb^$, one defines the corresponding directional derivative at a point $x\; \backslash in\; \backslash mathbb^$ by :$\backslash forall\; f\; \backslash in\; (\backslash mathbb^):\; \backslash qquad\; (D\_\; f)(x)\; ~\; \backslash stackrel\; ~\; \backslash left.\; \backslash frac\; [f(x\; +\; t\; v)]\; \backslash \_\; =\; \backslash sum\_^\; v^\; (x).$ This map is naturally a derivation at $x$. Furthermore, every derivation at a point in $\backslash mathbb^$ 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 :$\backslash forall\; f\; \backslash in\; (M):\; \backslash qquad\; (f)\; ~\; \backslash stackrel\; ~\; v(f).$ If we think of $v$ as the initial velocity of a differentiable curve $\backslash gamma$ initialized at $x$, i.e., $v\; =\; \backslash gamma\text{'}(0)$, then instead, define $D\_$ by :$\backslash forall\; f\; \backslash in\; (M):\; \backslash qquad\; (f)\; ~\; \backslash stackrel\; ~\; (f\; \backslash circ\; \backslash gamma)\text{'}(0).$Basis of the tangent space at a point

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

Every smooth (or differentiable) map $\backslash varphi:\; M\; \backslash to\; N$ between smooth (or differentiable) manifolds induces natural linear maps between their corresponding tangent spaces: :$\backslash mathrm\_:\; T\_\; M\; \backslash to\; T\_\; N.$ If the tangent space is defined via differentiable curves, then this map is defined by :$(\backslash gamma\text{'}(0))\; ~\; \backslash stackrel\; ~\; (\backslash varphi\; \backslash circ\; \backslash gamma)\text{'}(0).$ If, instead, the tangent space is defined via derivations, then this map is defined by :$[\backslash mathrm\_(D)](f)\; ~\; \backslash stackrel\; ~\; D(f\; \backslash circ\; \backslash varphi).$ The linear map $\backslash mathrm\_$ is called variously the ''derivative'', ''total derivative'', ''differential'', or ''pushforward'' of $\backslash varphi$ at $x$. It is frequently expressed using a variety of other notations: :$D\; \backslash varphi\_,\; \backslash qquad\; (\backslash varphi\_)\_,\; \backslash qquad\; \backslash varphi\text{'}(x).$ In a sense, the derivative is the best linear approximation to $\backslash varphi$ near $x$. Note that when $N\; =\; \backslash mathbb$, then the map $\backslash mathrm\_:\; T\_\; M\; \backslash to\; \backslash mathbb$ coincides with the usual notion of the Differential (calculus), differential of the function $\backslash varphi$. In local coordinates the derivative of $\backslash varphi$ is given by the Jacobian matrix and determinant, Jacobian. An important result regarding the derivative map is the following: :Theorem. If $\backslash varphi:\; M\; \backslash to\; N$ is a local diffeomorphism at $x$ in $M$, then $\backslash mathrm\_:\; T\_\; M\; \backslash to\; T\_\; N$ is a linear isomorphism. Conversely, if $\backslash mathrm\_$ is an isomorphism, then there is an open set, open neighborhood $U$ of $x$ such that $\backslash varphi$ maps $U$ diffeomorphically onto its image. This is a generalization of the inverse function theorem to maps between manifolds.See also

* Exponential map (Riemannian geometry), Exponential map * Vector space * Differential geometry of curves * Coordinate-induced basis * Cotangent spaceNotes

References

* . * . * .External links

Tangent Planes

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