In the

^{T} and ^{T} denote the matrix transpose, transpose of the vectors and , respectively. Under a change of basis of the form
:$\backslash mathbf\backslash mapsto\; \backslash mathbf\text{'}\; =\; \backslash left(\backslash sum\_k\; X\_ka\_,\backslash dots,\backslash sum\_k\; X\_ka\_\backslash right)\; =\; \backslash mathbfA$
for some invertible matrix, invertible matrix , the matrix of components of the metric changes by as well. That is,
:$G[\backslash mathbfA]\; =\; A^\backslash mathsf\; G[\backslash mathbf]A$
or, in terms of the entries of this matrix,
:$g\_[\backslash mathbfA]\; =\; \backslash sum\_^n\; a\_g\_[\backslash mathbf]a\_\; \backslash ,\; .$
For this reason, the system of quantities is said to transform covariantly with respect to changes in the frame .

^{T} the (covariant) components of the covector field , where
:$a\_i[\backslash mathbf]\; =\; \backslash sum\_^n\; v^k[\backslash mathbf]g\_[\backslash mathbf]$
is called lowering the index.
To ''raise the index'', one applies the same construction but with the inverse metric instead of the metric. If are the components of a covector in the dual basis , then the column vector
has components which transform contravariantly:
:$v[\backslash mathbfA]\; =\; A^v[\backslash mathbf].$
Consequently, the quantity does not depend on the choice of basis in an essential way, and thus defines a vector field on . The operation () associating to the (covariant) components of a covector the (contravariant) components of a vector given is called raising the index. In components, () is
:$v^i[\backslash mathbf]\; =\; \backslash sum\_^n\; g^[\backslash mathbf]\; a\_k[\backslash mathbf].$

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 (mathematics), dimensi ...

, the partial derivatives are orthonormal with respect to the Euclidean metric. Thus the metric tensor is the Kronecker delta δ_{''ij''} in this coordinate system. The metric tensor with respect to arbitrary (possibly curvilinear) coordinates is given by
:$g\_\; =\; \backslash sum\_\backslash delta\_\backslash frac\; \backslash frac\; =\; \backslash sum\_k\backslash frac\backslash frac.$

"Disquisitiones generales circa superficies curvas"

''Commentationes Societatis Regiae Scientiarum Gottingesis Recentiores'' Vol. VI (1827), pp. 99–146. * . * . * . * . * (''to appear''). * * * * * {{Manifolds Riemannian geometry Tensors Concepts in physics Differential geometry Metric tensors, *1

mathematical
Mathematics (from Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximately 10.7 ...

field of 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 geometry of curves, theor ...

, one definition of a metric tensor is a type of function which takes as input a pair of 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 and at a point of a surface (or higher dimensional 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 ...

) and produces a real number
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). ...

scalar
Scalar may refer to:
*Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers
*Scalar (physics), a physical quantity that can be described by a single element of a number field such as ...

in a way that generalizes many of the familiar properties of the dot product
In mathematics, the dot product or scalar productThe term ''scalar product'' is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space. is an algebraic operation that takes two equal-length seque ...

of vectors
Vector may refer to:
Biology
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism; a disease vector
*Vector (molecular biology), a DNA molecule used as a vehicle to artificially carr ...

in 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 (mathematics), dimensi ...

. In the same way as a dot product, metric tensors are used to define the length of and angle between tangent vectors. Through , the metric tensor allows one to define and compute the length of curves on the manifold.
A metric tensor is called ''positive-definite'' if it assigns a positive value to every nonzero vector . A manifold equipped with a positive-definite metric tensor is known as a Riemannian 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 Differentia ...

. On a Riemannian manifold, the curve connecting two points that (locally) has the smallest length is called a geodesic
In 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 properties of space t ...

, and its length is the distance that a passenger in the manifold needs to traverse to go from one point to the other. Equipped with this notion of length, a Riemannian manifold is a metric 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). It ...

, meaning that it has a distance function
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 ...

whose value at a pair of points and is the distance from to . Conversely, the metric tensor itself is the derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its Argument of a function, argument (input value). Derivatives are a fundament ...

of the distance function (taken in a suitable manner). Thus the metric tensor gives the ''infinitesimal'' distance on the manifold.
While the notion of a metric tensor was known in some sense to mathematicians such as Carl Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes referr ...

from the early 19th century, it was not until the early 20th century that its properties as a tensor
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). ...

were understood by, in particular, Gregorio Ricci-Curbastro
Gregorio Ricci-Curbastro (; 12January 1925) was an Italian mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity ( ...

and Tullio Levi-Civita
Tullio Levi-Civita, (, ; 29 March 1873 – 29 December 1941) was an Italians, Italian mathematician, most famous for his work on absolute differential calculus (tensor calculus) and its applications to the theory of relativity, but who also made s ...

, who first codified the notion of a tensor. The metric tensor is an example of a tensor field
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 ...

.
The components of a metric tensor in a coordinate basisIn 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 ...

take on the form of a symmetric matrix
In linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces an ...

whose entries transform covariantly under changes to the coordinate system. Thus a metric tensor is a covariant symmetric tensor
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 ...

. From the coordinate-independent point of view, a metric tensor field is defined to be a nondegenerate symmetric bilinear formA symmetric bilinear form on a vector space is a bilinear map from two copies of the vector space to the field of scalars such that the order of the two vectors does not affect the value of the map. In other words, it is a bilinear function B tha ...

on each tangent space that varies from point to point.
Introduction

Carl Friedrich Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician
This is a List of German mathematician
A mathematician is someone who uses an extensive knowledge of ma ...

in his 1827 '' Disquisitiones generales circa superficies curvas'' (''General investigations of curved surfaces'') considered a surface parametrically, with the Cartesian coordinates
A Cartesian coordinate system (, ) in a plane
Plane or planes may refer to:
* Airplane or aeroplane or informally plane, a powered, fixed-wing aircraft
Arts, entertainment and media
*Plane (Dungeons & Dragons), Plane (''Dungeons & Dragons'') ...

, , and of points on the surface depending on two auxiliary variables and . Thus a parametric surface is (in today's terms) a vector-valued function
A vector-valued function, also referred to as a vector function, is a function (mathematics), mathematical function of one or more variables whose range of a function, range is a set of multidimensional Euclidean vector, vectors or infinite-dimensi ...

:$\backslash vec(u,\backslash ,v)\; =\; \backslash bigl(\; x(u,\backslash ,v),\backslash ,\; y(u,\backslash ,v),\backslash ,\; z(u,\backslash ,v)\; \backslash bigr)$
depending on an ordered pair
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 real variables , and defined in an open set
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 ...

in the -plane. One of the chief aims of Gauss's investigations was to deduce those features of the surface which could be described by a function which would remain unchanged if the surface underwent a transformation in space (such as bending the surface without stretching it), or a change in the particular parametric form of the same geometrical surface.
One natural such invariant quantity is the drawn along the surface. Another is the angle
In Euclidean geometry, an angle is the figure formed by two Ray (geometry), rays, called the ''sides'' of the angle, sharing a common endpoint, called the ''vertex (geometry), vertex'' of the angle.
Angles formed by two rays lie in the plane (ge ...

between a pair of curves drawn along the surface and meeting at a common point. A third such quantity is the area
Area is the quantity that expresses the extent of a two-dimensional region, shape, or planar lamina, in the plane. Surface area is its analog on the two-dimensional surface of a three-dimensional object. Area can be understood as the am ...

of a piece of the surface. The study of these invariants of a surface led Gauss to introduce the predecessor of the modern notion of the metric tensor.
Arc length

If the variables and are taken to depend on a third variable, , taking values in an interval , then will trace out aparametric curve
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 ...

in parametric surface . The arc length
Arc length is the distance between two points along a section 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.
Int ...

of that curve is given by the integral
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). ...

: $\backslash begin\; s\; \&=\; \backslash int\_a^b\backslash left\backslash ,\; \backslash frac\backslash vec(u(t),v(t))\backslash right\backslash ,\; \backslash ,dt\; \backslash \backslash $ &= \int_a^b \sqrt\, dt \,,
\end
where $\backslash left\backslash ,\; \backslash cdot\; \backslash right\backslash ,$ represents the Euclidean norm
Euclidean space is the fundamental space of classical geometry. Originally it was the three-dimensional space
Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called pa ...

. Here the chain rule has been applied, and the subscripts denote partial derivative
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 ...

s:
:$\backslash vec\_u\; =\; \backslash frac\backslash ,,\; \backslash quad\; \backslash vec\_v\; =\; \backslash frac\backslash ,.$
The integrand is the restriction to the curve of the square root of the ( quadratic) differential
where
The quantity in () is called the line element
In 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 properties of space tha ...

, while is called the first fundamental form
In differential geometry, the first fundamental form is the inner product on the tangent space of a surface (differential geometry), surface in three-dimensional Euclidean space which is induced canonical form, canonically from the dot product of . ...

of . Intuitively, it represents the principal part
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 the square of the displacement undergone by when is increased by units, and is increased by units.
Using matrix notation, the first fundamental form becomes
:$ds^2\; =\; \backslash begin\; du\; \&\; dv\; \backslash end\; \backslash begin\; E\; \&\; F\; \backslash \backslash \; F\; \&\; G\; \backslash end\; \backslash begin\; du\; \backslash \backslash \; dv\; \backslash end$
Coordinate transformations

Suppose now that a different parameterization is selected, by allowing and to depend on another pair of variables and . Then the analog of () for the new variables is The chain rule relates , , and to , , and via thematrix
Matrix or MATRIX may refer to:
Science and mathematics
* Matrix (mathematics), a rectangular array of numbers, symbols, or expressions
* Matrix (logic), part of a formula in prenex normal form
* Matrix (biology), the material in between a eukaryoti ...

equation
where the superscript T denotes the matrix transpose
In linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and ...

. The matrix with the coefficients , , and arranged in this way therefore transforms by the Jacobian matrix
In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix (mathematics), matrix of all its first-order partial derivatives. When this matrix is square matrix, square, that is, when the function ta ...

of the coordinate change
:$J\; =\; \backslash begin\; \backslash frac\; \&\; \backslash frac\; \backslash \backslash \; \backslash frac\; \&\; \backslash frac\; \backslash end\backslash ,.$
A matrix which transforms in this way is one kind of what is called a tensor
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). ...

. The matrix
:$\backslash begin\; E\; \&\; F\; \backslash \backslash \; F\; \&\; G\; \backslash end$
with the transformation law () is known as the metric tensor of the surface.
Invariance of arclength under coordinate transformations

first observed the significance of a system of coefficients , , and , that transformed in this way on passing from one system of coordinates to another. The upshot is that the first fundamental form () is ''invariant'' under changes in the coordinate system, and that this follows exclusively from the transformation properties of , , and . Indeed, by the chain rule, :$\backslash begin\; du\; \backslash \backslash \; dv\; \backslash end\; =\; \backslash begin\; \backslash dfrac\; \&\; \backslash dfrac\; \backslash \backslash \; \backslash dfrac\; \&\; \backslash dfrac\; \backslash end\; \backslash begin\; du\text{'}\; \backslash \backslash \; dv\text{'}\; \backslash end$ so that :$\backslash begin\; ds^2\; \&=\; \backslash begin\; du\; \&\; dv\; \backslash end\; \backslash begin\; E\; \&\; F\; \backslash \backslash \; F\; \&\; G\; \backslash end\; \backslash begin\; du\; \backslash \backslash \; dv\; \backslash end\; \backslash \backslash $ &= \begin du' & dv' \end \begin \dfrac & \dfrac \\ \dfrac & \dfrac \end^\mathsf \begin E & F \\ F & G \end \begin \dfrac & \dfrac \\ \dfrac & \dfrac \end \begin du' \\ dv' \end \\ &= \begin du' & dv' \end \begin E' & F' \\ F' & G' \end \begin du' \\ dv' \end\\ &= (ds')^2 \,. \endLength and angle

Another interpretation of the metric tensor, also considered by Gauss, is that it provides a way in which to compute the length oftangent 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 to the surface, as well as the angle between two tangent vectors. In contemporary terms, the metric tensor allows one to compute the dot product
In mathematics, the dot product or scalar productThe term ''scalar product'' is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space. is an algebraic operation that takes two equal-length seque ...

of tangent vectors in a manner independent of the parametric description of the surface. Any tangent vector at a point of the parametric surface can be written in the form
:$\backslash mathbf\; =\; p\_1\backslash vec\_u\; +\; p\_2\backslash vec\_v$
for suitable real numbers and . If two tangent vectors are given:
:$\backslash begin\; \backslash mathbf\; \&=\; a\_1\backslash vec\_u\; +\; a\_2\backslash vec\_v\; \backslash \backslash \; \backslash mathbf\; \&=\; b\_1\backslash vec\_u\; +\; b\_2\backslash vec\_v\; \backslash end$
then using the bilinearity of the dot product,
:$\backslash begin\; \backslash mathbf\; \backslash cdot\; \backslash mathbf\; \&=\; a\_1\; b\_1\; \backslash vec\_u\backslash cdot\backslash vec\_u\; +\; a\_1b\_2\; \backslash vec\_u\backslash cdot\backslash vec\_v\; +\; b\_1a\_2\; \backslash vec\_v\backslash cdot\backslash vec\_u\; +\; a\_2\; b\_2\; \backslash vec\_v\backslash cdot\backslash vec\_v\; \backslash \backslash $ &= a_1 b_1 E + a_1b_2 F + b_1a_2 F + a_2b_2G \\ &= \begin a_1 & a_2 \end
\begin E & F \\ F & G \end
\begin b_1 \\ b_2 \end \,.
\end
This is plainly a function of the four variables , , , and . It is more profitably viewed, however, as a function that takes a pair of arguments and which are vectors in the -plane. That is, put
:$g(\backslash mathbf,\; \backslash mathbf)\; =\; a\_1b\_1\; E\; +\; a\_1b\_2\; F\; +\; b\_1a\_2\; F\; +\; a\_2b\_2G\; \backslash ,.$
This is a symmetric function
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 ...

in and , meaning that
:$g(\backslash mathbf,\; \backslash mathbf)\; =\; g(\backslash mathbf,\; \backslash mathbf)\backslash ,.$
It is also bilinear, meaning that it is linear
Linearity is the property of a mathematical relationship (''function
Function or functionality may refer to:
Computing
* Function key
A function key is a key on a computer
A computer is a machine that can be programmed to carry out se ...

in each variable and separately. That is,
:$\backslash begin\; g\backslash left(\backslash lambda\backslash mathbf\; +\; \backslash mu\backslash mathbf\text{'},\; \backslash mathbf\backslash right)\; \&=\; \backslash lambda\; g(\backslash mathbf,\; \backslash mathbf)\; +\; \backslash mu\; g\backslash left(\backslash mathbf\text{'},\; \backslash mathbf\backslash right),\backslash quad\backslash text\; \backslash \backslash \; g\backslash left(\backslash mathbf,\; \backslash lambda\backslash mathbf\; +\; \backslash mu\backslash mathbf\text{'}\backslash right)\; \&=\; \backslash lambda\; g(\backslash mathbf,\; \backslash mathbf)\; +\; \backslash mu\; g\backslash left(\backslash mathbf,\; \backslash mathbf\text{'}\backslash right)\; \backslash end$
for any vectors , , , and in the plane, and any real numbers and .
In particular, the length of a tangent vector is given by
:$\backslash left\backslash ,\; \backslash mathbf\; \backslash right\backslash ,\; =\; \backslash sqrt$
and the angle between two vectors and is calculated by
:$\backslash cos(\backslash theta)\; =\; \backslash frac\; \backslash ,.$
Area

Thesurface area
of radius has surface area .
The surface area of a Solid geometry, solid object is a measure of the total area
Area is the quantity that expresses the extent of a two-dimensional region, shape, or planar lamina, in the plane. Surface ...

is another numerical quantity which should depend only on the surface itself, and not on how it is parameterized. If the surface is parameterized by the function over the domain in the -plane, then the surface area of is given by the integral
:$\backslash iint\_D\; \backslash left,\; \backslash vec\_u\; \backslash times\; \backslash vec\_v\backslash \backslash ,du\backslash ,dv$
where denotes the cross 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 the absolute value denotes the length of a vector in Euclidean space. By Lagrange's identity
In algebra
Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. I ...

for the cross product, the integral can be written
:$\backslash begin\; \&\backslash iint\_D\; \backslash sqrt\backslash ,du\backslash ,dv\; \backslash \backslash $ = &\iint_D \sqrt\,du\,dv\\ = &\iint_D \sqrt\, du\, dv
\end
where is the determinant
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 ...

.
Definition

Let be asmooth 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 a ...

of dimension ; for instance a surface
File:Water droplet lying on a damask.jpg, Water droplet lying on a damask. Surface tension is high enough to prevent floating below the textile.
A surface, as the term is most generally used, is the outermost or uppermost layer of a physical obje ...

(in the case ) or hypersurface
In 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 properties of space that ...

in the Cartesian spaceCartesian means of or relating to the French philosopher René Descartes
René Descartes ( or ; ; Latinized: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a French-born philosopher, mathematician
A mathematician is someone wh ...

. At each point there is a 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). ...

, called the tangent space
In mathematics, the tangent space of a manifold facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector that gives the Displacement (vector ...

, consisting of all tangent vectors to the manifold at the point . A metric tensor at is a function which takes as inputs a pair of tangent vectors and at , and produces as an output a real number
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). ...

(scalar
Scalar may refer to:
*Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers
*Scalar (physics), a physical quantity that can be described by a single element of a number field such as ...

), so that the following conditions are satisfied:
* is bilinear. A function of two vector arguments is bilinear if it is linear separately in each argument. Thus if , , are three tangent vectors at and and are real numbers, then
*:$\backslash begin\; g\_p(aU\_p\; +\; bV\_p,\; Y\_p)\; \&=\; ag\_p(U\_p,\; Y\_p)\; +\; bg\_p(V\_p,\; Y\_p)\; \backslash ,,\; \backslash quad\; \backslash text\; \backslash \backslash \; g\_p(Y\_p,\; aU\_p\; +\; bV\_p)\; \&=\; ag\_p(Y\_p,\; U\_p)\; +\; bg\_p(Y\_p,\; V\_p)\; \backslash ,.\; \backslash end$
* is symmetric
Symmetry (from Greek συμμετρία ''symmetria'' "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more pre ...

. A function of two vector arguments is symmetric provided that for all vectors and ,
*:$g\_p(X\_p,\; Y\_p)\; =\; g\_p(Y\_p,\; X\_p)\backslash ,.$
* is nondegenerate. A bilinear function is nondegenerate provided that, for every tangent vector , the function
*:$Y\_p\backslash mapsto\; g\_p(X\_p,Y\_p)$
:obtained by holding constant and allowing to vary is not identically zero. That is, for every there exists a such that .
A metric tensor field on assigns to each point of a metric tensor in the tangent space at in a way that varies with . More precisely, given any open set, open subset of manifold and any (smooth) vector fields and on , the real function
:$g(X,\; Y)(p)\; =\; g\_p(X\_p,\; Y\_p)$
is a smooth function of .
Components of the metric

The components of the metric in any basis of a vector space, basis of vector fields, or frame bundle, frame, are given by The functions form the entries of ansymmetric matrix
In linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces an ...

, . If
:$v\; =\; \backslash sum\_^n\; v^iX\_i\; \backslash ,,\; \backslash quad\; w\; =\; \backslash sum\_^n\; w^iX\_i$
are two vectors at , then the value of the metric applied to and is determined by the coefficients () by bilinearity:
:$g(v,\; w)\; =\; \backslash sum\_^n\; v^iw^jg\backslash left(X\_i,X\_j\backslash right)\; =\; \backslash sum\_^n\; v^iw^jg\_[\backslash mathbf]$
Denoting the matrix
Matrix or MATRIX may refer to:
Science and mathematics
* Matrix (mathematics), a rectangular array of numbers, symbols, or expressions
* Matrix (logic), part of a formula in prenex normal form
* Matrix (biology), the material in between a eukaryoti ...

by and arranging the components of the vectors and into column vectors and ,
:$g(v,w)\; =\; \backslash mathbf[\backslash mathbf]^\backslash mathsf\; G[\backslash mathbf]\; \backslash mathbf[\backslash mathbf]\; =\; \backslash mathbf[\backslash mathbf]^\backslash mathsf\; G[\backslash mathbf]\backslash mathbf[\backslash mathbf]$
where Metric in coordinates

A system of real-valued functions , giving a local coordinates, local coordinate system on anopen set
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 ...

in , determines a basis of vector fields on
:$\backslash mathbf\; =\; \backslash left(X\_1\; =\; \backslash frac,\; \backslash dots,\; X\_n\; =\; \backslash frac\backslash right)\; \backslash ,.$
The metric has components relative to this frame given by
:$g\_\backslash left[\backslash mathbf\backslash right]\; =\; g\backslash left(\backslash frac,\; \backslash frac\backslash right)\; \backslash ,.$
Relative to a new system of local coordinates, say
:$y^i\; =\; y^i(x^1,\; x^2,\; \backslash dots,\; x^n),\backslash quad\; i=1,2,\backslash dots,n$
the metric tensor will determine a different matrix of coefficients,
:$g\_\backslash left[\backslash mathbf\text{'}\backslash right]\; =\; g\backslash left(\backslash frac,\; \backslash frac\backslash right).$
This new system of functions is related to the original by means of the chain rule
:$\backslash frac\; =\; \backslash sum\_^n\; \backslash frac\backslash frac$
so that
:$g\_\backslash left[\backslash mathbf\text{'}\backslash right]\; =\; \backslash sum\_^n\; \backslash frac\; g\_\backslash left[\backslash mathbf\backslash right]\backslash frac.$
Or, in terms of the matrices and ,
:$G\backslash left[\backslash mathbf\text{'}\backslash right]\; =\; \backslash left((Dy)^\backslash right)^\backslash mathsf\; G\backslash left[\backslash mathbf\backslash right]\; (Dy)^$
where denotes the Jacobian matrix
In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix (mathematics), matrix of all its first-order partial derivatives. When this matrix is square matrix, square, that is, when the function ta ...

of the coordinate change.
Signature of a metric

Associated to any metric tensor is the quadratic form defined in each tangent space by :$q\_m(X\_m)\; =\; g\_m(X\_m,X\_m)\; \backslash ,,\; \backslash quad\; X\_m\backslash in\; T\_mM.$ If is positive for all non-zero , then the metric is definite bilinear form, positive-definite at . If the metric is positive-definite at every , then is called a Riemannian metric. More generally, if the quadratic forms have constant signature of a quadratic form, signature independent of , then the signature of is this signature, and is called a pseudo-Riemannian metric. If is connected space, connected, then the signature of does not depend on . By Sylvester's law of inertia, a basis of tangent vectors can be chosen locally so that the quadratic form diagonalizes in the following manner :$q\_m\backslash left(\backslash sum\_i\backslash xi^iX\_i\backslash right)\; =\; \backslash left(\backslash xi^1\backslash right)^2+\backslash left(\backslash xi^2\backslash right)^2+\backslash cdots+\backslash left(\backslash xi^p\backslash right)^2\; -\; \backslash left(\backslash xi^\backslash right)^2-\backslash cdots-\backslash left(\backslash xi^n\backslash right)^2$ for some between 1 and . Any two such expressions of (at the same point of ) will have the same number of positive signs. The signature of is the pair of integers , signifying that there are positive signs and negative signs in any such expression. Equivalently, the metric has signature if the matrix of the metric has positive and negative eigenvalues. Certain metric signatures which arise frequently in applications are: * If has signature , then is a Riemannian metric, and is called aRiemannian 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 Differentia ...

. Otherwise, is a pseudo-Riemannian metric, and is called a pseudo-Riemannian manifold (the term semi-Riemannian is also used).
* If is four-dimensional with signature or , then the metric is called Lorentzian metric, Lorentzian. More generally, a metric tensor in dimension other than 4 of signature or is sometimes also called Lorentzian.
* If is -dimensional and has signature , then the metric is called ultrahyperbolic metric, ultrahyperbolic.
Inverse metric

Let be a basis of vector fields, and as above let be the matrix of coefficients :$g\_[\backslash mathbf]\; =\; g\backslash left(X\_i,X\_j\backslash right)\; \backslash ,.$ One can consider the inverse matrix , which is identified with the inverse metric (or ''conjugate'' or ''dual metric''). The inverse metric satisfies a transformation law when the frame is changed by a matrix via The inverse metric transforms ''Covariance and contravariance of vectors, contravariantly'', or with respect to the inverse of the change of basis matrix . Whereas the metric itself provides a way to measure the length of (or angle between) vector fields, the inverse metric supplies a means of measuring the length of (or angle between) covector fields; that is, fields of linear functionals. To see this, suppose that is a covector field. To wit, for each point , determines a function defined on tangent vectors at so that the following linear transformation, linearity condition holds for all tangent vectors and , and all real numbers and : :$\backslash alpha\_p\; \backslash left(aX\_p\; +\; bY\_p\backslash right)\; =\; a\backslash alpha\_p\; \backslash left(X\_p\backslash right)\; +\; b\backslash alpha\_p\; \backslash left(Y\_p\backslash right)\backslash ,.$ As varies, is assumed to be a smooth function in the sense that :$p\; \backslash mapsto\; \backslash alpha\_p\; \backslash left(X\_p\backslash right)$ is a smooth function of for any smooth vector field . Any covector field has components in the basis of vector fields . These are determined by :$\backslash alpha\_i\; =\; \backslash alpha\; \backslash left(X\_i\backslash right)\backslash ,,\backslash quad\; i\; =\; 1,\; 2,\; \backslash dots,\; n\backslash ,.$ Denote the row vector of these components by :$\backslash alpha[\backslash mathbf]\; =\; \backslash big\backslash lbrack\backslash begin\; \backslash alpha\_1\; \&\; \backslash alpha\_2\; \&\; \backslash dots\; \&\; \backslash alpha\_n\; \backslash end\backslash big\backslash rbrack\; \backslash ,.$ Under a change of by a matrix , changes by the rule :$\backslash alpha[\backslash mathbfA]\; =\; \backslash alpha[\backslash mathbf]A\; \backslash ,.$ That is, the row vector of components transforms as a ''covariant'' vector. For a pair and of covector fields, define the inverse metric applied to these two covectors by The resulting definition, although it involves the choice of basis , does not actually depend on in an essential way. Indeed, changing basis to gives :$\backslash begin\; \&\backslash alpha[\backslash mathbfA]\; G[\backslash mathbfA]^\; \backslash beta[\backslash mathbfA]^\backslash mathsf\; \backslash \backslash \; =\; \&\backslash left(\backslash alpha[\backslash mathbf]A\backslash right)\; \backslash left(A^G[\backslash mathbf]^\; \backslash left(A^\backslash right)^\backslash mathsf\backslash right)\; \backslash left(A^\backslash mathsf\backslash beta[\backslash mathbf]^\backslash mathsf\backslash right)\; \backslash \backslash \; =\; \&\backslash alpha[\backslash mathbf]\; G[\backslash mathbf]^\; \backslash beta[\backslash mathbf]^\backslash mathsf.\; \backslash end$ So that the right-hand side of equation () is unaffected by changing the basis to any other basis whatsoever. Consequently, the equation may be assigned a meaning independently of the choice of basis. The entries of the matrix are denoted by , where the indices and have been raised to indicate the transformation law ().Raising and lowering indices

In a basis of vector fields , any smooth tangent vector field can be written in the form for some uniquely determined smooth functions . Upon changing the basis by a nonsingular matrix , the coefficients change in such a way that equation () remains true. That is, :$X\; =\; \backslash mathbfv[\backslash mathbf]\; =\; \backslash mathbfv[\backslash mathbf]\backslash ,.$ Consequently, . In other words, the components of a vector transform ''contravariantly'' (that is, inversely or in the opposite way) under a change of basis by the nonsingular matrix . The contravariance of the components of is notationally designated by placing the indices of in the upper position. A frame also allows covectors to be expressed in terms of their components. For the basis of vector fields define the dual basis to be the linear functionals such that :$\backslash theta^i[\backslash mathbf](X\_j)\; =\; \backslash begin\; 1\; \&\; \backslash mathrm\backslash \; i=j\backslash \backslash \; 0\&\backslash mathrm\backslash \; i\backslash not=j.\backslash end$ That is, , the Kronecker delta. Let :$\backslash theta[\backslash mathbf]\; =\; \backslash begin\backslash theta^1[\backslash mathbf]\; \backslash \backslash \; \backslash theta^2[\backslash mathbf]\; \backslash \backslash \; \backslash vdots\; \backslash \backslash \; \backslash theta^n[\backslash mathbf]\backslash end.$ Under a change of basis for a nonsingular matrix , transforms via :$\backslash theta[\backslash mathbfA]\; =\; A^\backslash theta[\backslash mathbf].$ Any linear functional on tangent vectors can be expanded in terms of the dual basis where denotes the row vector . The components transform when the basis is replaced by in such a way that equation () continues to hold. That is, :$\backslash alpha\; =\; a[\backslash mathbfA]\backslash theta[\backslash mathbfA]\; =\; a[\backslash mathbf]\backslash theta[\backslash mathbf]$ whence, because , it follows that . That is, the components transform ''covariantly'' (by the matrix rather than its inverse). The covariance of the components of is notationally designated by placing the indices of in the lower position. Now, the metric tensor gives a means to identify vectors and covectors as follows. Holding fixed, the function :$g\_p(X\_p,\; -)\; :\; Y\_p\; \backslash mapsto\; g\_p(X\_p,\; Y\_p)$ of tangent vector defines a linear functional on the tangent space at . This operation takes a vector at a point and produces a covector . In a basis of vector fields , if a vector field has components , then the components of the covector field in the dual basis are given by the entries of the row vector :$a[\backslash mathbf]\; =\; v[\backslash mathbf]^\backslash mathsf\; G[\backslash mathbf].$ Under a change of basis , the right-hand side of this equation transforms via :$v[\backslash mathbfA]^\backslash mathsf\; G[\backslash mathbfA]\; =\; v[\backslash mathbf]^\backslash mathsf\; \backslash left(A^\backslash right)^\backslash mathsf\; A^\backslash mathsf\; G[\backslash mathbf]A\; =\; v[\backslash mathbf]^\backslash mathsf\; G[\backslash mathbf]A$ so that : transforms covariantly. The operation of associating to the (contravariant) components of a vector fieldInduced metric

Let be anopen set
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 ...

in , and let be a continuously differentiable function from into the 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 (mathematics), dimensi ...

, where . The mapping is called an immersion (mathematics), immersion if its differential is injective at every point of . The image of is called an immersed submanifold. More specifically, for , which means that the ambient 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 (mathematics), dimensi ...

is , the induced metric tensor is called the First_fundamental_form, first fundamental form.
Suppose that is an immersion onto the submanifold . The usual Euclidean dot product
In mathematics, the dot product or scalar productThe term ''scalar product'' is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space. is an algebraic operation that takes two equal-length seque ...

in is a metric which, when restricted to vectors tangent to , gives a means for taking the dot product of these tangent vectors. This is called the induced metric.
Suppose that is a tangent vector at a point of , say
:$v\; =\; v^1\backslash mathbf\_1\; +\; \backslash dots\; +\; v^n\backslash mathbf\_n$
where are the standard coordinate vectors in . When is applied to , the vector goes over to the vector tangent to given by
:$\backslash varphi\_*(v)\; =\; \backslash sum\_^n\; \backslash sum\_^m\; v^i\backslash frac\backslash mathbf\_a\backslash ,.$
(This is called the pushforward (differential), pushforward of along .) Given two such vectors, and , the induced metric is defined by
:$g(v,w)\; =\; \backslash varphi\_*(v)\backslash cdot\; \backslash varphi\_*(w).$
It follows from a straightforward calculation that the matrix of the induced metric in the basis of coordinate vector fields is given by
:$G(\backslash mathbf)\; =\; (D\backslash varphi)^\backslash mathsf(D\backslash varphi)$
where is the Jacobian matrix:
:$D\backslash varphi\; =\; \backslash begin\; \backslash frac\; \&\; \backslash frac\; \&\; \backslash dots\; \&\; \backslash frac\; \backslash \backslash [1ex]\; \backslash frac\; \&\; \backslash frac\; \&\; \backslash dots\; \&\; \backslash frac\; \backslash \backslash \; \backslash vdots\; \&\; \backslash vdots\; \&\; \backslash ddots\; \&\; \backslash vdots\; \backslash \backslash \; \backslash frac\; \&\; \backslash frac\; \&\; \backslash dots\; \&\; \backslash frac\; \backslash end.$
Intrinsic definitions of a metric

The notion of a metric can be defined intrinsically using the language of fiber bundles and vector bundles. In these terms, a metric tensor is a function from the fiber product of the tangent bundle of with itself to such that the restriction of to each fiber is a nondegenerate bilinear mapping :$g\_p\; :\; \backslash mathrm\_pM\backslash times\; \backslash mathrm\_pM\; \backslash to\; \backslash mathbf.$ The mapping () is required to be continuous function, continuous, and often continuously differentiable, smooth function, smooth, or real analytic, depending on the case of interest, and whether can support such a structure.Metric as a section of a bundle

By the Tensor product#Universal property, universal property of the tensor product, any bilinear mapping () gives rise natural transformation, naturally to a section (fiber bundle), section of the dual space, dual of the tensor product bundle of with itself :$g\_\backslash otimes\; \backslash in\; \backslash Gamma\backslash left((\backslash mathrmM\; \backslash otimes\; \backslash mathrmM)^*\backslash right).$ The section is defined on simple elements of by :$g\_\backslash otimes(v\; \backslash otimes\; w)\; =\; g(v,\; w)$ and is defined on arbitrary elements of by extending linearly to linear combinations of simple elements. The original bilinear form is symmetric if and only if :$g\_\backslash otimes\; \backslash circ\; \backslash tau\; =\; g\_\backslash otimes$ where :$\backslash tau\; :\; \backslash mathrmM\; \backslash otimes\; \backslash mathrmM\; \backslash stackrel\; TM\; \backslash otimes\; TM$ is the tensor product#Tensor powers and braiding, braiding map. Since is finite-dimensional, there is a natural isomorphism :$(\backslash mathrmM\; \backslash otimes\; \backslash mathrmM)^*\; \backslash cong\; \backslash mathrm^*M\; \backslash otimes\; \backslash mathrm^*M,$ so that is regarded also as a section of the bundle of the cotangent bundle with itself. Since is symmetric as a bilinear mapping, it follows that is asymmetric tensor
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 ...

.
Metric in a vector bundle

More generally, one may speak of a metric in a vector bundle. If is a vector bundle over a manifold , then a metric is a mapping :$g\; :\; E\backslash times\_M\; E\backslash to\; \backslash mathbf$ from the fiber product of to which is bilinear in each fiber: :$g\_p\; :\; E\_p\; \backslash times\; E\_p\backslash to\; \backslash mathbf.$ Using duality as above, a metric is often identified with a section (fiber bundle), section of the tensor product bundle . (See metric (vector bundle).)Tangent–cotangent isomorphism

The metric tensor gives a natural isomorphism from the tangent bundle to the cotangent bundle, sometimes called the musical isomorphism. This isomorphism is obtained by setting, for each tangent vector , :$S\_gX\_p\backslash ,\; \backslash stackrel\backslash text\backslash ,\; g(X\_p,\; -),$ the linear functional on which sends a tangent vector at to . That is, in terms of the pairing between and its dual space , :$[S\_gX\_p,\; Y\_p]\; =\; g\_p(X\_p,\; Y\_p)$ for all tangent vectors and . The mapping is a linear transformation from to . It follows from the definition of non-degeneracy that the kernel (set theory), kernel of is reduced to zero, and so by the rank–nullity theorem, is a linear isomorphism. Furthermore, is a symmetric linear transformation in the sense that :$[S\_gX\_p,\; Y\_p]\; =\; [S\_gY\_p,\; X\_p]$ for all tangent vectors and . Conversely, any linear isomorphism defines a non-degenerate bilinear form on by means of :$g\_S(X\_p,\; Y\_p)\; =\; [SX\_p,\; Y\_p]\backslash ,.$ This bilinear form is symmetric if and only if is symmetric. There is thus a natural one-to-one correspondence between symmetric bilinear forms on and symmetric linear isomorphisms of to the dual . As varies over , defines a section of the bundle of vector bundle morphism, vector bundle isomorphisms of the tangent bundle to the cotangent bundle. This section has the same smoothness as : it is continuous, differentiable, smooth, or real-analytic according as . The mapping , which associates to every vector field on a covector field on gives an abstract formulation of "lowering the index" on a vector field. The inverse of is a mapping which, analogously, gives an abstract formulation of "raising the index" on a covector field. The inverse defines a linear mapping :$S\_g^\; :\; \backslash mathrm^*M\; \backslash to\; \backslash mathrmM$ which is nonsingular and symmetric in the sense that :$\backslash left[S\_g^\backslash alpha,\; \backslash beta\backslash right]\; =\; \backslash left[S\_g^\backslash beta,\; \backslash alpha\backslash right]$ for all covectors , . Such a nonsingular symmetric mapping gives rise (by the tensor-hom adjunction) to a map :$\backslash mathrm^*M\; \backslash otimes\; \backslash mathrm^*M\; \backslash to\; \backslash mathbf$ or by the Double dual, double dual isomorphism to a section of the tensor product :$\backslash mathrmM\; \backslash otimes\; \backslash mathrmM.$Arclength and the line element

Suppose that is a Riemannian metric on . In a local coordinate system , , the metric tensor appears as a matrix (math), matrix, denoted here by , whose entries are the components of the metric tensor relative to the coordinate vector fields. Let be a piecewise-differentiableparametric curve
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 ...

in , for . The arclength of the curve is defined by
:$L\; =\; \backslash int\_a^b\; \backslash sqrt\backslash ,dt\; \backslash ,.$
In connection with this geometrical application, the quadratic differential form
:$ds^2\; =\; \backslash sum\_^n\; g\_(p)\; dx^i\; dx^j$
is called the first fundamental form
In differential geometry, the first fundamental form is the inner product on the tangent space of a surface (differential geometry), surface in three-dimensional Euclidean space which is induced canonical form, canonically from the dot product of . ...

associated to the metric, while is the line element
In 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 properties of space tha ...

. When is pullback (differential geometry), pulled back to the image of a curve in , it represents the square of the differential with respect to arclength.
For a pseudo-Riemannian metric, the length formula above is not always defined, because the term under the square root may become negative. We generally only define the length of a curve when the quantity under the square root is always of one sign or the other. In this case, define
:$L\; =\; \backslash int\_a^b\; \backslash sqrt\backslash ,dt\; \backslash ,\; .$
Note that, while these formulas use coordinate expressions, they are in fact independent of the coordinates chosen; they depend only on the metric, and the curve along which the formula is integrated.
The energy, variational principles and geodesics

Given a segment of a curve, another frequently defined quantity is the (kinetic) energy of the curve: :$E\; =\; \backslash frac\; \backslash int\_a^b\; \backslash sum\_^ng\_(\backslash gamma(t))\; \backslash left(\backslash fracx^i\; \backslash circ\; \backslash gamma(t)\backslash right)\backslash left(\backslash fracx^j\; \backslash circ\; \backslash gamma(t)\backslash right)\backslash ,dt\; \backslash ,.$ This usage comes from physics, specifically, classical mechanics, where the integral can be seen to directly correspond to the kinetic energy of a point particle moving on the surface of a manifold. Thus, for example, in Jacobi's formulation of Maupertuis' principle, the metric tensor can be seen to correspond to the mass tensor of a moving particle. In many cases, whenever a calculation calls for the length to be used, a similar calculation using the energy may be done as well. This often leads to simpler formulas by avoiding the need for the square-root. Thus, for example, the geodesic equations may be obtained by applying variational principles to either the length or the energy. In the latter case, the geodesic equations are seen to arise from the principle of least action: they describe the motion of a "free particle" (a particle feeling no forces) that is confined to move on the manifold, but otherwise moves freely, with constant momentum, within the manifold.Canonical measure and volume form

In analogy with the case of surfaces, a metric tensor on an -dimensional paracompact manifold gives rise to a natural way to measure the -dimensional volume of subsets of the manifold. The resulting natural positive Borel measure allows one to develop a theory of integrating functions on the manifold by means of the associated Lebesgue integral. A measure can be defined, by the Riesz representation theorem, by giving a positive linear functional on the space of compact support, compactly supported continuous functions on . More precisely, if is a manifold with a (pseudo-)Riemannian metric tensor , then there is a unique positive Borel measure such that for any coordinate chart , :$\backslash Lambda\; f\; =\; \backslash int\_U\; f\backslash ,d\backslash mu\_g\; =\; \backslash int\_\; f\backslash circ\backslash varphi^(x)\; \backslash sqrt\backslash ,dx$ for all supported in . Here is thedeterminant
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 ...

of the matrix formed by the components of the metric tensor in the coordinate chart. That is well-defined on functions supported in coordinate neighborhoods is justified by integration by substitution, Jacobian change of variables. It extends to a unique positive linear functional on by means of a partition of unity.
If is also orientation (mathematics), oriented, then it is possible to define a natural volume form from the metric tensor. In a right-handed coordinate system, positively oriented coordinate system the volume form is represented as
:$\backslash omega\; =\; \backslash sqrt\backslash ,\; dx^1\backslash wedge\backslash cdots\backslash wedge\; dx^n$
where the are the coordinate differentials and denotes the exterior product in the algebra of differential forms. The volume form also gives a way to integrate functions on the manifold, and this geometric integral agrees with the integral obtained by the canonical Borel measure.
Examples

Euclidean metric

The most familiar example is that of elementary Euclidean geometry: the two-dimensional Euclidean distance, Euclidean metric tensor. In the usual coordinates, we can write :$g\; =\; \backslash begin\; 1\; \&\; 0\; \backslash \backslash \; 0\; \&\; 1\backslash end\; \backslash ,.$ The length of a curve reduces to the formula: :$L\; =\; \backslash int\_a^b\; \backslash sqrt\; \backslash ,.$ The Euclidean metric in some other common coordinate systems can be written as follows. Polar coordinates : :$\backslash begin\; x\; \&=\; r\; \backslash cos\backslash theta\; \backslash \backslash \; y\; \&=\; r\; \backslash sin\backslash theta\; \backslash \backslash \; J\; \&=\; \backslash begin\backslash cos\backslash theta\; \&\; -r\backslash sin\backslash theta\; \backslash \backslash \; \backslash sin\backslash theta\; \&\; r\backslash cos\backslash theta\backslash end\; \backslash ,.\; \backslash end$ So :$g\; =\; J^\backslash mathsfJ\; =\; \backslash begin\; \backslash cos^2\backslash theta\; +\; \backslash sin^2\backslash theta\; \&\; -r\backslash sin\backslash theta\; \backslash cos\backslash theta\; +\; r\backslash sin\backslash theta\backslash cos\backslash theta\; \backslash \backslash \; -r\backslash cos\backslash theta\backslash sin\backslash theta\; +\; r\backslash cos\backslash theta\backslash sin\backslash theta\; \&\; r^2\; \backslash sin^2\backslash theta\; +\; r^2\backslash cos^2\backslash theta\; \backslash end\; =\; \backslash begin\; 1\; \&\; 0\; \backslash \backslash \; 0\; \&\; r^2\; \backslash end$ by trigonometric identity, trigonometric identities. In general, in a Cartesian coordinate system on aThe round metric on a sphere

The unit sphere in comes equipped with a natural metric induced from the ambient Euclidean metric, through the process explained in the Metric_tensor#Induced_metric, induced metric section. In standard spherical coordinates , with the colatitude, the angle measured from the -axis, and the angle from the -axis in the -plane, the metric takes the form :$g\; =\; \backslash begin\; 1\; \&\; 0\; \backslash \backslash \; 0\; \&\; \backslash sin^2\; \backslash theta\backslash end\; \backslash ,.$ This is usually written in the form :$ds^2\; =\; d\backslash theta^2\; +\; \backslash sin^2\backslash theta\backslash ,d\backslash varphi^2\backslash ,.$Lorentzian metrics from relativity

In flat Minkowski space (special relativity), with coordinates :$r^\backslash mu\; \backslash rightarrow\; \backslash left(x^0,\; x^1,\; x^2,\; x^3\backslash right)\; =\; (ct,\; x,\; y,\; z)\; \backslash ,\; ,$ the metric is, depending on choice of metric signature, :$g\; =\; \backslash begin\; 1\; \&\; 0\; \&\; 0\; \&\; 0\backslash \backslash \; 0\; \&\; -1\; \&\; 0\; \&\; 0\; \backslash \backslash \; 0\; \&\; 0\; \&\; -1\; \&\; 0\; \backslash \backslash \; 0\; \&\; 0\; \&\; 0\; \&\; -1\; \backslash end\; \backslash quad\; \backslash text\; \backslash quad\; g\; =\; \backslash begin\; -1\; \&\; 0\; \&\; 0\; \&\; 0\backslash \backslash \; 0\; \&\; 1\; \&\; 0\; \&\; 0\; \backslash \backslash \; 0\; \&\; 0\; \&\; 1\; \&\; 0\; \backslash \backslash \; 0\; \&\; 0\; \&\; 0\; \&\; 1\; \backslash end\; \backslash ,.$ For a curve with—for example—constant time coordinate, the length formula with this metric reduces to the usual length formula. For a Spacetime interval, timelike curve, the length formula gives the proper time along the curve. In this case, the spacetime interval is written as :$ds^2\; =\; c^2\; dt^2\; -\; dx^2\; -\; dy^2\; -\; dz^2\; =\; dr^\backslash mu\; dr\_\backslash mu\; =\; g\_\; dr^\backslash mu\; dr^\backslash nu\backslash ,.$ The Schwarzschild metric describes the spacetime around a spherically symmetric body, such as a planet, or a black hole. With coordinates :$\backslash left(x^0,\; x^1,\; x^2,\; x^3\backslash right)\; =\; (ct,\; r,\; \backslash theta,\; \backslash varphi)\; \backslash ,,$ we can write the metric as :$g\_\; =\; \backslash begin\; \backslash left(1\; -\; \backslash frac\backslash right)\; \&\; 0\; \&\; 0\; \&\; 0\; \backslash \backslash \; 0\; \&\; -\backslash left(1\; -\; \backslash frac\backslash right)^\; \&\; 0\; \&\; 0\; \backslash \backslash \; 0\; \&\; 0\; \&\; -r^2\; \&\; 0\; \backslash \backslash \; 0\; \&\; 0\; \&\; 0\; \&\; -r^2\; \backslash sin^2\; \backslash theta\; \backslash end\backslash ,,$ where (inside the matrix) is the gravitational constant and represents the total mass-energy content of the central object.See also

* Basic introduction to the mathematics of curved spacetime * Clifford algebra * Finsler manifold * List of coordinate charts * Ricci calculus * Tissot's indicatrix, a technique to visualize the metric tensorNotes

References

* * . * translated by A. M. Hiltebeitel and J. C. Morehead"Disquisitiones generales circa superficies curvas"

''Commentationes Societatis Regiae Scientiarum Gottingesis Recentiores'' Vol. VI (1827), pp. 99–146. * . * . * . * . * (''to appear''). * * * * * {{Manifolds Riemannian geometry Tensors Concepts in physics Differential geometry Metric tensors, *1