In

Tangent Planes

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

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

Indifferential 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 $\backslash mathbb^$ 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 acurve
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 " ...

$k\; \backslash geq\; 1$) and that $x\; \backslash in\; M$. 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 ...

$\backslash varphi:\; U\; \backslash to\; \backslash mathbb^$, 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 $\backslash gamma\_,\backslash gamma\_:\; (-\; 1,1)\; \backslash to\; M$ 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
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 $\backslash gamma$ 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 $\backslash varphi:\; U\; \backslash to\; \backslash mathbb^$.
To define vector-space operations on $T\_\; M$, we use a chart $\backslash varphi:\; U\; \backslash to\; \backslash mathbb^$ 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 ...

$\backslash mathrm\_:\; T\_\; M\; \backslash to\; \backslash mathbb^$ by $(\backslash gamma\text{\'}(0))\; \backslash mathrel\; \backslash left.\; \backslash frac;\; href="/html/ALL/s/\backslash varphi\_\backslash circ\_\backslash gamma)(t).html"\; ;"title="\backslash varphi\; \backslash circ\; \backslash gamma)(t)">\backslash varphi\; \backslash circ\; \backslash gamma)(t)$ 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
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 $\backslash mathbb^$ 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 $f\; \backslash circ\; \backslash varphi^:\; \backslash varphi;\; href="/html/ALL/s/.html"\; ;"title="">$ is infinitely differentiable. Note that $(M)$ is a realassociative 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 $x\; \backslash in\; M$. 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 ...

$D:\; (M)\; \backslash to\; \backslash mathbb$ that satisfies the Leibniz identity
$$\backslash forall\; f,g\; \backslash in\; (M):\; \backslash qquad\; D(f\; g)\; =\; D(f)\; \backslash cdot\; g(x)\; +\; f(x)\; \backslash cdot\; D(g),$$
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 $f=\backslash text,$ it follows that $D(f)=0$).
If we define addition and scalar multiplication on the set of derivations at $x$ by
* $(D\_1+D\_2)(f)\; \backslash mathrel\; \_1(f)\; +\; \_2(f)$ and
* $(\backslash lambda\; \backslash cdot\; D)(f)\; \backslash mathrel\; \backslash lambda\; \backslash cdot\; D(f)$,
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, tocomplex 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 ...

$\backslash mathcal\_$. 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 $p\; \backslash in\; X$ is the collection of all $\backslash mathbb$-derivations $D:\; \backslash mathcal\_\; \backslash to\; \backslash mathbb$, where $\backslash mathbb$ 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 $\backslash mathcal\_$ 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 $\backslash mathcal\_$ 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 $(f)\; \backslash mathrel\; (f\; \backslash circ\; \backslash gamma)\text{'}(0)$ (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 $(f)\; \backslash mathrel\; (f\; \backslash circ\; \backslash gamma)\text{'}(0),$ 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 theideal
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 $C^\backslash infty(M)$ 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 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 $D(f)\; =\; 0$ 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 $D(f)\; \backslash mathrel\; r\backslash left((f\; -\; f(x))\; +\; I^2\backslash right)$ 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 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 asdirectional 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 $\backslash mathbb^$, one defines the corresponding directional derivative at a point $x\; \backslash in\; \backslash mathbb^$ by
:$\backslash forall\; f\; \backslash in\; (\backslash mathbb^):\; \backslash qquad\; (D\_\; f)(x)\; \backslash mathrel\; \backslash left.\; \backslash frac;\; href="/html/ALL/s/(x\_+\_t\_v).html"\; ;"title="(x\; +\; t\; v)">(x\; +\; t\; v)$
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
:$\backslash forall\; f\; \backslash in\; (M):\; \backslash qquad\; (f)\; \backslash mathrel\; v(f).$
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
:$\backslash forall\; f\; \backslash in\; (M):\; \backslash qquad\; (f)\; \backslash mathrel\; (f\; \backslash circ\; \backslash gamma)\text{'}(0).$
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 :$\backslash forall\; i\; \backslash in\; \backslash ,\; ~\; \backslash forall\; f\; \backslash in\; (M):\; \backslash qquad\; (f)\; \backslash mathrel\; \backslash left(\; \backslash frac\; \backslash Big(\; f\; \backslash circ\; \backslash varphi^\; \backslash Big)\; \backslash right)\; \backslash Big(\; \backslash varphi(p)\; \backslash Big)\; .$ 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 naturallinear 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:
:$\backslash mathrm\_:\; T\_\; M\; \backslash to\; T\_\; N.$
If the tangent space is defined via differentiable curves, then this map is defined by
:$(\backslash gamma\text{'}(0))\; \backslash mathrel\; (\backslash varphi\; \backslash circ\; \backslash gamma)\text{'}(0).$
If, instead, the tangent space is defined via derivations, then this map is defined by
:$;\; href="/html/ALL/s/mathrm\_(D).html"\; ;"title="mathrm\_(D)">mathrm\_(D)$
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 of the function $\backslash varphi$. 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 $\backslash varphi$ 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.
See also

* 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 ...

Notes

References

* . * . * .External links

Tangent Planes

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