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In
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. 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 ...
, 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 {{Redirect category shell, 1= {{R for alternate capitalisation ...
{{Redirect category shell, 1= {{R for alternate capitalisation ...
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
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 $\mathbb^$ 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 $k \geq 1$) and that $x \in M$. Pick a Manifold#Charts, atlases, and transition maps, coordinate chart $\varphi: U \to \mathbb^$, where $U$ is an open set, open subset of $M$ containing $x$. Suppose further that two curves $\gamma_,\gamma_: \left(- 1,1\right) \to M$ with $\left(0\right) = x = \left(0\right)$ are given such that both $\varphi \circ \gamma_,\varphi \circ \gamma_: \left(- 1,1\right) \to \mathbb^$ 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 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 $\gamma$ is denoted by $\gamma\text{'}\left(0\right)$. 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 $\varphi: U \to \mathbb^$. Image:Tangentialvektor.svg, left, 200px, The tangent space $T_ M$ and a tangent vector $v \in T_ M$, along a curve traveling through $x \in M$. To define vector-space operations on $T_ M$, we use a chart $\varphi: U \to \mathbb^$ and define a Map (mathematics), map $\mathrm_: T_ M \to \mathbb^$ by $\left(\gamma\text{'}\left(0\right)\right) ~ \stackrel ~ \left. \frac \left[\left(\varphi \circ \gamma\right)\left(t\right)\right] \_,$ where $\gamma \in \gamma\text{'}\left(0\right)$. 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. The map $\mathrm_$ turns out to be bijective 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.

## Definition via derivations

Suppose now that $M$ is a $C^$ manifold. A real-valued function $f: M \to \mathbb$ is said to belong to $\left(M\right)$ if and only if for every coordinate chart $\varphi: U \to \mathbb^$, the map $f \circ \varphi^: \varphi\left[U\right] \subseteq \mathbb^ \to \mathbb$ is infinitely differentiable. Note that $\left(M\right)$ is a real associative algebra with respect to the pointwise product and sum of functions and scalar multiplication. Pick a point $x \in M$. A ''Derivation (abstract algebra), derivation'' at $x$ is defined as a linear map $D: \left(M\right) \to \mathbb$ that satisfies the Leibniz identity :$\forall f,g \in \left(M\right): \qquad D\left(f g\right) = D\left(f\right) \cdot g + f \cdot D\left(g\right),$ which is modeled on the product rule of calculus. (For every identically constant function $f=\text,$ it follows that $D\left(f\right)=0$). If we define addition and scalar multiplication on the set of derivations at $x$ by * $\left(D_1+D_2\right)\left(f\right) ~ \stackrel ~ _1\left(f\right) + _2\left(f\right)$ and * $\left(\lambda \cdot D\right)\left(f\right) ~ \stackrel ~ \lambda \cdot D\left(f\right)$, 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 $\mathcal_$. 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, and ...
at a point $p \in X$ is the collection of all $\mathbb$-derivations $D: \mathcal_ \to \mathbb$, where $\mathbb$ is the ground field and $\mathcal_$ is the stalk (sheaf), stalk of $\mathcal_$ at $p$.

## Equivalence of the definitions

For $x \in M$ and a differentiable curve $\gamma: \left(- 1,1\right) \to M$ such that $\gamma \left(0\right) = x,$ define $\left(f\right) ~ \stackrel ~ \left(f \circ \gamma\right)\text{'}\left(0\right)$ (where the derivative is taken in the ordinary sense because $f \circ \gamma$ is a function from $\left(- 1,1\right)$ to $\mathbb$). One can ascertain that $D_\left(f\right)$ is a derivation at the point $x,$ and that equivalent curves yield the same derivation. Thus, for an equivalence class $\gamma\text{'}\left(0\right),$ we can define $\left(f\right) \stackrel \left(f \circ \gamma\right)\text{'}\left(0\right),$ where the curve $\gamma \in \gamma\text{'}\left(0\right)$ has been chosen arbitrarily. The map $\gamma\text{'}\left(0\right) \mapsto D_$ is a vector space isomorphism between the space of the equivalence classes $\gamma\text{'}\left(0\right)$ 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 (ring theory), ideal $I$ of $C^\infty\left(M\right)$ that consists of all smooth functions $f$ vanishing at $x$, i.e., $f\left(x\right) = 0$. 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 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 ...
. If $D$ is a derivation at $x$, then $D\left(f\right) = 0$ 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 $D\left(f\right) ~ \stackrel ~ r\left\left(\left(f - f\left(x\right)\right) + I^2\right\right)$ 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 function, 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 derivatives. Given a vector $v$ in $\mathbb^$, one defines the corresponding directional derivative at a point $x \in \mathbb^$ by :$\forall f \in \left(\mathbb^\right): \qquad \left(D_ f\right)\left(x\right) ~ \stackrel ~ \left. \frac \left[f\left(x + t v\right)\right] \_ = \sum_^ v^ \left(x\right).$ 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 :$\forall f \in \left(M\right): \qquad \left(f\right) ~ \stackrel ~ v\left(f\right).$ If we think of $v$ as the initial velocity of a differentiable curve $\gamma$ initialized at $x$, i.e., $v = \gamma\text{'}\left(0\right)$, then instead, define $D_$ by :$\forall f \in \left(M\right): \qquad \left(f\right) ~ \stackrel ~ \left(f \circ \gamma\right)\text{'}\left(0\right).$

## Basis of the tangent space at a point

For a $C^$ manifold $M$, if a chart $\varphi = \left(x^,\ldots,x^\right): U \to \mathbb^$ is given with $p \in U$, then one can define an ordered basis $\left\$ of $T_ M$ by :$\forall i \in \, ~ \forall f \in \left(M\right): \qquad \left(f\right) ~ \stackrel ~ \left\left( \frac \Big\left( f \circ \varphi^ \Big\right) \right\right) \Big\left( \varphi\left(p\right) \Big\right) .$ Then for every tangent vector $v \in T_ M$, one has :$v = \sum_^ v^ \cdot \left\left( \frac \right\right)_.$ This formula therefore expresses $v$ as a linear combination of the basis tangent vectors $\left\left( \frac \right\right)_ \in T_ M$ 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 maps 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 :$\left(\gamma\text{'}\left(0\right)\right) ~ \stackrel ~ \left(\varphi \circ \gamma\right)\text{'}\left(0\right).$ If, instead, the tangent space is defined via derivations, then this map is defined by :$\left[\mathrm_\left(D\right)\right]\left(f\right) ~ \stackrel ~ D\left(f \circ \varphi\right).$ 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: :$D \varphi_, \qquad \left(\varphi_\right)_, \qquad \varphi\text{'}\left(x\right).$ 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 (calculus), differential of the function $\varphi$. In local coordinates the derivative of $\varphi$ is given by the Jacobian matrix and determinant, Jacobian. An important result regarding the derivative map is the following: :Theorem. If $\varphi: M \to N$ is a local diffeomorphism at $x$ in $M$, then $\mathrm_: T_ M \to T_ N$ is a linear isomorphism. Conversely, if $\mathrm_$ is an isomorphism, then there is an open set, open neighborhood $U$ of $x$ such that $\varphi$ maps $U$ diffeomorphically onto its image. This is a generalization of the inverse function theorem to maps between manifolds.

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

* . * . * .