In
mathematics, the tangent space of a
manifold generalizes to higher dimensions the notion of ''
tangent planes'' to surfaces in three dimensions and ''
tangent line
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...
s'' to curves in two dimensions. In the context of physics the tangent space to a manifold at a point can be viewed as the space of possible velocities for a particle moving on the manifold.
Informal description
In
differential geometry, one can attach to every point
of a
differentiable manifold a ''tangent space''—a real
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
that intuitively contains the possible directions in which one can tangentially pass through
. The elements of the tangent space at
are called the ''
tangent vector
In mathematics, a tangent vector is a vector that is tangent to a curve or surface at a given point. Tangent vectors are described in the differential geometry of curves in the context of curves in R''n''. More generally, tangent vectors are e ...
s'' at
. This is a generalization of the notion of a
vector
Vector most often refers to:
*Euclidean vector, a quantity with a magnitude and a direction
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematic ...
, based at a given initial point, in a
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
. The
dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...
of the tangent space at every point of a
connected
Connected may refer to:
Film and television
* ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular''
* '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film
* ''Connected'' (2015 TV ...
manifold is the same as that of the
manifold itself.
For example, if the given manifold is a
-
sphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is th ...
, then one can picture the tangent space at a point as the plane that touches the sphere at that point and is
perpendicular
In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It ca ...
to the sphere's radius through the point. More generally, if a given manifold is thought of as an
embedded submanifold
In mathematics, a submanifold of a manifold ''M'' is a subset ''S'' which itself has the structure of a manifold, and for which the inclusion map satisfies certain properties. There are different types of submanifolds depending on exactly which ...
of
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
, then one can picture a tangent space in this literal fashion. This was the traditional approach toward defining
parallel transport
In geometry, parallel transport (or parallel translation) is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection (a covariant derivative or connection on the tangent b ...
. Many authors in
differential geometry and
general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
use it. More strictly, this defines an affine tangent space, which is distinct from the space of tangent vectors described by modern terminology.
In
algebraic geometry, 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. ...
that gives a vector space with dimension at least that of
itself. The points
at which the dimension of the tangent space is exactly that of
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
are those where the "test to be a manifold" fails. See
Zariski tangent space
In algebraic geometry, the Zariski tangent space is a construction that defines a tangent space at a point ''P'' on an algebraic variety ''V'' (and more generally). It does not use differential calculus, being based directly on abstract algebra, an ...
.
Once the tangent spaces of a manifold have been introduced, one can define
vector fields, which are abstractions of the velocity field of particles moving in space. A vector field attaches to every point of the manifold a vector from the tangent space at that point, in a smooth manner. Such a vector field serves to define a generalized
ordinary differential equation
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast ...
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, but that does not have to be straight.
Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
on the manifold whose derivative at any point is equal to the tangent vector attached to that point by the vector field.
All the tangent spaces of a manifold may be "glued together" to form a new differentiable manifold with twice the dimension of the original manifold, called the ''
tangent bundle
In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of ...
'' of the manifold.
Formal definitions
The informal description above relies on a manifold's ability to be embedded into an ambient vector space
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
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, but that does not have to be straight.
Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
passing through the point
. We can therefore define a tangent vector as an equivalence class of curves passing through
while being tangent to each other at
.
Suppose that
is a
differentiable manifold (with
smoothness ) and that
. Pick a
coordinate chart In topology, a branch of mathematics, a topological manifold is a topological space that locally resembles real ''n''-dimensional Euclidean space. Topological manifolds are an important class of topological spaces, with applications throughout mathe ...
, where
is an
open subset of
containing
. Suppose further that two curves
with
are given such that both
are differentiable in the ordinary sense (we call these ''differentiable curves initialized at
''). Then
and
are said to be ''equivalent'' at
if and only if the derivatives of
and
at
coincide. This defines an
equivalence relation on the set of all differentiable curves initialized at
, and
equivalence classes of such curves are known as ''tangent vectors'' of
at
. The equivalence class of any such curve
is denoted by
. The ''tangent space'' of
at
, denoted by
, is then defined as the set of all tangent vectors at
; it does not depend on the choice of coordinate chart
.
To define vector-space operations on
, we use a chart
and define a
map
A map is a symbolic depiction emphasizing relationships between elements of some space, such as objects, regions, or themes.
Many maps are static, fixed to paper or some other durable medium, while others are dynamic or interactive. Although ...
by
where
. The map
turns out to be
bijective
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
and may be used to transfer the vector-space operations on
over to
, thus turning the latter set into an
-dimensional real vector space. Again, one needs to check that this construction does not depend on the particular chart
and the curve
being used, and in fact it does not.
Definition via derivations
Suppose now that
is a
manifold. A real-valued function
is said to belong to
if and only if for every coordinate chart
, the map
is infinitely differentiable. Note that
is a real
associative algebra with respect to the
pointwise product and sum of functions and scalar multiplication.
A ''
derivation
Derivation may refer to:
Language
* Morphological derivation, a word-formation process
* Parse tree or concrete syntax tree, representing a string's syntax in formal grammars
Law
* Derivative work, in copyright law
* Derivation proceeding, a proc ...
'' at
is defined as a
linear map
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
that satisfies the Leibniz identity
which is modeled on the
product rule
In calculus, the product rule (or Leibniz rule or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions. For two functions, it may be stated in Lagrange's notation as (u \cdot v)' = u ' \cdot v ...
of calculus.
(For every identically constant function
it follows that
).
Denote
the set of all derivations at
Setting
*
and
*
turns
into a vector space.
Generalizations
Generalizations of this definition are possible, for instance, to
complex manifolds and
algebraic varieties
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...
. However, instead of examining derivations
from the full algebra of functions, one must instead work at the level of
germs of functions. The reason for this is that the
structure sheaf may not be
fine for such structures. For example, let
be an algebraic variety with
structure sheaf . Then the
Zariski tangent space
In algebraic geometry, the Zariski tangent space is a construction that defines a tangent space at a point ''P'' on an algebraic variety ''V'' (and more generally). It does not use differential calculus, being based directly on abstract algebra, an ...
at a point
is the collection of all
-derivations
, where
is the
ground field In mathematics, a ground field is a field ''K'' fixed at the beginning of the discussion.
Use
It is used in various areas of algebra:
In linear algebra
In linear algebra, the concept of a vector space may be developed over any field.
In algeb ...
and
is the
stalk of
at
.
Equivalence of the definitions
For
and a differentiable curve
such that
define
(where the derivative is taken in the ordinary sense because
is a function from
to
). One can ascertain that
is a derivation at the point
and that equivalent curves yield the same derivation. Thus, for an equivalence class
we can define
where the curve
has been chosen arbitrarily. The map
is a vector space isomorphism between the space of the equivalence classes
and that of the derivations at the point
Definition via cotangent spaces
Again, we start with a
manifold
and a point
. Consider the
ideal
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considere ...
of
that consists of all smooth functions
vanishing at
, i.e.,
. Then
and
are both real vector spaces, and the
quotient space can be shown to be
isomorphic to the
cotangent space
In differential geometry, the cotangent space is a vector space associated with a point x on a smooth (or differentiable) manifold \mathcal M; one can define a cotangent space for every point on a smooth manifold. Typically, the cotangent space, T ...
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 th ...
. The tangent space
may then be defined as the
dual space of
.
While this definition is the most abstract, it is also the one that is most easily transferable to other settings, for instance, to the
varieties
Variety may refer to:
Arts and entertainment Entertainment formats
* Variety (radio)
* Variety show, in theater and television
Films
* ''Variety'' (1925 film), a German silent film directed by Ewald Andre Dupont
* ''Variety'' (1935 film), ...
considered in
algebraic geometry.
If
is a derivation at
, then
for every
, which means that
gives rise to a linear map
. Conversely, if
is a linear map, then
defines a derivation at
. This yields an equivalence between tangent spaces defined via derivations and tangent spaces defined via cotangent spaces.
Properties
If
is an open subset of
, then
is a
manifold in a natural manner (take coordinate charts to be
identity maps on open subsets of
), and the tangent spaces are all naturally identified with
.
Tangent vectors as directional derivatives
Another way to think about tangent vectors is as
directional derivative
In mathematics, the directional derivative of a multivariable differentiable (scalar) function along a given vector v at a given point x intuitively represents the instantaneous rate of change of the function, moving through x with a velocity ...
s. Given a vector
in
, one defines the corresponding directional derivative at a point
by
:
This map is naturally a derivation at
. Furthermore, every derivation at a point in
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
is a tangent vector to
at a point
(thought of as a derivation), then define the directional derivative
in the direction
by
:
If we think of
as the initial velocity of a differentiable curve
initialized at
, i.e.,
, then instead, define
by
:
Basis of the tangent space at a point
For a
manifold
, if a chart
is given with
, then one can define an ordered basis
of
by
:
Then for every tangent vector
, one has
:
This formula therefore expresses
as a linear combination of the basis tangent vectors
defined by the coordinate chart
.
The derivative of a map
Every smooth (or differentiable) map
between smooth (or differentiable) manifolds induces natural
linear map
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
s between their corresponding tangent spaces:
:
If the tangent space is defined via differentiable curves, then this map is defined by
:
If, instead, the tangent space is defined via derivations, then this map is defined by
:
The linear map
is called variously the ''derivative'', ''total derivative'', ''differential'', or ''pushforward'' of
at
. It is frequently expressed using a variety of other notations:
:
In a sense, the derivative is the best linear approximation to
near
. Note that when
, then the map
coincides with the usual notion of the
differential of the function
. In
local coordinates
Local coordinates are the ones used in a ''local coordinate system'' or a ''local coordinate space''. Simple examples:
* Houses. In order to work in a house construction, the measurements are referred to a control arbitrary point that will allow ...
the derivative of
is given by the
Jacobian.
An important result regarding the derivative map is the following:
This is a generalization of the
inverse function theorem
In mathematics, specifically differential calculus, the inverse function theorem gives a sufficient condition for a function to be invertible in a neighborhood of a point in its domain: namely, that its ''derivative is continuous and non-zero at ...
to maps between manifolds.
See also
*
Coordinate-induced basis
*
Cotangent space
In differential geometry, the cotangent space is a vector space associated with a point x on a smooth (or differentiable) manifold \mathcal M; one can define a cotangent space for every point on a smooth manifold. Typically, the cotangent space, T ...
*
Differential geometry of curves
Differential geometry of curves is the branch of geometry that deals with smooth curves in the plane and the Euclidean space by methods of differential and integral calculus.
Many specific curves have been thoroughly investigated using the ...
*
Exponential map
*
Vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
Notes
References
* .
* .
* .
External links
Tangent Planesat MathWorld
{{DEFAULTSORT:Tangent Space
Differential topology
Differential geometry