TheInfoList

In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, 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 Mathematics (from Ancient Greek, Greek: ) includes the study of su ...

generalizes to higher dimensions the notion of tangent planes to surfaces in three dimensions and tangent lines 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 Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds, using the techniques of differential calculus, integral calculus, linear algebra a ...
, one can attach to every point $x$ of a
differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of 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 surfa ...
a ''tangent space''—a real
vector space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
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 :''For a more general — but much more technical — treatment of tangent vectors, see tangent space.'' In mathematics, a tangent vector is a Vector (geometry), vector that is tangent to a curve or Surface (mathematics), surface at a given point. T ...
s'' at $x$. This is a generalization of the notion of a
vector Vector may refer to: Biology *Vector (epidemiology) In epidemiology Epidemiology is the study and analysis of the distribution (who, when, and where), patterns and risk factor, determinants of health and disease conditions in defined pop ...
, based at a given initial point, in a
Euclidean space Euclidean space is the fundamental space of . Originally, it was the of , but in modern there are Euclidean spaces of any nonnegative integer , including the three-dimensional space and the ''Euclidean plane'' (dimension two). It was introduce ...
. The
dimension In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...
of the tangent space at every point of a connected 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 Mathematics (from Ancient Greek, Greek: ) includes the study of su ...

itself. For example, if the given manifold is a $2$-
sphere A sphere (from Greek language, Greek —, "globe, ball") is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a circle in two-dimensional space. A sphere is the Locus (mathematics), set of points that are ...

, 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 Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with propertie ...

to the sphere's radius through the point. More generally, if a given manifold is thought of as an embedded
submanifold In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
of
Euclidean space Euclidean space is the fundamental space of . Originally, it was the of , but in modern there are Euclidean spaces of any nonnegative integer , including the three-dimensional space and the ''Euclidean plane'' (dimension two). It was introduce ...
, then one can picture a tangent space in this literal fashion. This was the traditional approach toward defining
parallel transport In , parallel transport (or parallel translation) is a way of transporting geometrical data along smooth curves in a . If the manifold is equipped with an (a or on the ), then this connection allows one to transport vectors of the manifold al ...

. Many authors in
differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds, using the techniques of differential calculus, integral calculus, linear algebra a ...
and
general relativity General relativity, also known as the general theory of relativity, is the geometric Geometry (from the grc, γεωμετρία; '' geo-'' "earth", '' -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathema ...
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 Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and thei ...

, 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 Algebraic geometry is a branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematic ...
$V$ 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 Algebraic geometry is a branch of mathematics, classically studying zero of a function, zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commuta ...
. Once the tangent spaces of a manifold have been introduced, one can define
vector field In vector calculus Vector calculus, or vector analysis, is concerned with differentiation Differentiation may refer to: Business * Differentiation (economics), the process of making a product different from other similar products * Product ...

s, 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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
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 (geo ...

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 Image:Tangent bundle.svg, Informally, the tangent bundle of a manifold (which in this case is a circle) is obtained by considering all the tangent spaces (top), and joining them together in a smooth and non-overlapping manner (bottom). In differen ...

'' 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 (geo ...

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 The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surfa ...
(with
smoothness is a smooth function with compact support. In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain. At the very minimum, a function could be considered " ...
$k \geq 1$) and that $x \in M$. Pick a
coordinate chartIn topology, a branch of mathematics, a topological manifold is a topological space which locally resembles real numbers, real ''n''-dimension (mathematics), dimensional Euclidean space. Topological manifolds are an important class of topological spa ...
$\varphi: U \to \mathbb^$, where $U$ is an
open subset Open or OPEN may refer to: Music * Open (band) Open is a band. Background Drummer Pete Neville has been involved in the Sydney/Australian music scene for a number of years. He has recently completed a Masters in screen music at the Australian ...
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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
on the set of all differentiable curves initialized at $x$, and
equivalence class In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
es of such curves are known as ''tangent vectors'' of $M$ at $x$. The equivalence class of any such curve $\gamma$ is denoted by $\gamma\text{'}\left(0\right)$. 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 $\varphi: U \to \mathbb^$. To define vector-space operations on $T_ M$, we use a chart $\varphi: U \to \mathbb^$ and define a
map A map is a symbol A symbol is a mark, sign, or that indicates, signifies, or is understood as representing an , , or . Symbols allow people to go beyond what is n or seen by creating linkages between otherwise very different s and s. A ...
$\mathrm_: T_ M \to \mathbb^$ by 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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
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 is infinitely differentiable. Note that $\left(M\right)$ is a real
associative algebra In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
with respect to the
pointwise product In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
and sum of functions and scalar multiplication. Pick a point $x \in M$. A ''
derivation Derivation may refer to: * Derivation (differential algebra), a unary function satisfying the Leibniz product law * Derivation (linguistics) * Formal proof or derivation, a sequence of sentences each of which is an axiom or follows from the precedi ...
'' at $x$ is defined as a
linear map In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

$D: \left(M\right) \to \mathbb$ 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 (mathematics), functions. For two functions, it may be stated in Notation for differentiatio ...
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 * $\left(D_1+D_2\right)\left(f\right) \mathrel _1\left(f\right) + _2\left(f\right)$ and * $\left(\lambda \cdot D\right)\left(f\right) \mathrel \lambda \cdot D\left(f\right)$, 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 manifold In differential geometry Differential geometry is a Mathematics, mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The Differential ...
s and
algebraic varieties Algebraic varieties are the central objects of study in algebraic geometry Algebraic geometry is a branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures ...
. However, instead of examining derivations $D$ from the full algebra of functions, one must instead work at the level of
germs Germ or germs may refer to: Science * Germ (microorganism), an informal word for a pathogen * Germ cell, cell that gives rise to the gametes of an organism that reproduces sexually * Germ layer, a primary layer of cells that forms during embryon ...
of functions. The reason for this is that the
structure sheafIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
may not be
fine Fine may refer to: Characters * Sylvia Fine (''The Nanny''), Fran's mother on ''The Nanny'' * Fine/Officer Fine, character in Tales from the Crypt, played by Vincent Spano Vincent M. Spano (born October 18, 1962) is an American film, stage and ...
for such structures. For example, let $X$ be an algebraic variety with
structure sheafIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
$\mathcal_$. Then the
Zariski tangent space In algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zero of a function, zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commuta ...
at a point $p \in X$ is the collection of all $\mathbb$-derivations $D: \mathcal_ \to \mathbb$, where $\mathbb$ is the
ground fieldIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
and $\mathcal_$ is the
stalk Stalk or stalking may refer to: Behaviour * Stalk, the stealthy approach (phase) of a predator towards its prey * Stalking, an act of intrusive behaviour or unwanted attention towards a person * Deer stalking, the pursuit of deer for sport Biology ...
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 $\left(f\right) \mathrel \left(f \circ \gamma\right)\text{'}\left(0\right)$ (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 $\left(f\right) \mathrel \left(f \circ \gamma\right)\text{'}\left(0\right),$ 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 Ideal may refer to: Philosophy * Ideal (ethics) An ideal is a principle A principle is a proposition or value that is a guide for behavior or evaluation. In law Law is a system A system is a group of Interaction, interacting ...
$I$ 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 both real vector spaces, and the quotient space $I / I^2$ can be shown to be to the
cotangent space In differential geometry Differential geometry is a Mathematics, mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The Differential ...
$T^_x M$ 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 Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
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: Science and technology Mathematics * Algebraic variety Algebraic varieties are the central objects of study in algebraic geometry Algebraic geometry is a branch of mathematics Mathematics (from Ancient Greek, ...
considered in
algebraic geometry Algebraic geometry is a branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and thei ...

. 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 $D\left(f\right) \mathrel r\left\left(\left(f - f\left(x\right)\right) + I^2\right\right)$ 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 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 multivariate differentiable function, differentiable (scalar) function along a given vector (mathematics), vector v at a given point x intuitively represents the instantaneous rate of change of the ...
s. Given a vector $v$ in $\mathbb^$, one defines the corresponding directional derivative at a point $x \in \mathbb^$ by : 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 :$\forall f \in \left(M\right): \qquad \left(f\right) \mathrel v\left(f\right).$ 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 :$\forall f \in \left(M\right): \qquad \left(f\right) \mathrel \left(f \circ \gamma\right)\text{'}\left(0\right).$

## 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 :$\forall i \in \, ~ \forall f \in \left(M\right): \qquad \left(f\right) \mathrel \left\left( \frac \Big\left( f \circ \varphi^ \Big\right) \right\right) \Big\left( \varphi\left(p\right) \Big\right) .$ 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 $\left( \frac \right)_ \in T_ M$ 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 map In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

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 :$\left(\gamma\text{'}\left(0\right)\right) \mathrel \left(\varphi \circ \gamma\right)\text{'}\left(0\right).$ If, instead, the tangent space is defined via derivations, then this map is defined by : 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 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 t ...
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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
to maps between manifolds.

* Coordinate-induced basis *
Cotangent space In differential geometry Differential geometry is a Mathematics, mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The Differential ...
*
Differential geometry of curves Differential geometry of curves is the branch of geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematic ...
* Exponential map *
Vector space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

* . * . * .