Editing Metric tensor (section)

==Arclength and the line element==
Suppose that {{mvar|g}} is a Riemannian metric on {{mvar|M}}. In a local coordinate system {{math|''x''<sup>''i''</sup>}}, {{math|''i'' {{=}} 1, 2, …, ''n''}}, the metric tensor appears as a [[matrix (math)|matrix]], denoted here by {{math|'''G'''}}, whose entries are the components {{math|''g''<sub>''ij''</sub>}} of the metric tensor relative to the coordinate vector fields.

Let {{math|''γ''(''t'')}} be a piecewise-differentiable [[parametric curve]] in {{mvar|M}}, for {{math|''a'' ≤ ''t'' ≤ ''b''}}. The [[arclength]] of the curve is defined by

:<math>L = \int_a^b \sqrt{ \sum_{i,j=1}^n g_{ij}(\gamma(t)) \left(\frac{d}{dt}x^i \circ \gamma(t)\right) \left(\frac{d}{dt} x^j \circ \gamma(t)\right)}\,dt \,.</math>

In connection with this geometrical application, the [[quadratic form|quadratic]] [[differential form]]

:<math>ds^2 = \sum_{i,j=1}^n g_{ij}(p) dx^i dx^j</math>

is called the [[first fundamental form]] associated to the metric, while {{mvar|ds}} is the [[line element]]. When {{math|''ds''<sup>2</sup>}} is [[pullback (differential geometry)|pulled back]] to the image of a curve in {{mvar|M}}, 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

:<math>L = \int_a^b \sqrt{ \left|\sum_{i,j=1}^ng_{ij}(\gamma(t)) \left(\frac{d}{dt}x^i \circ \gamma(t)\right)\left(\frac{d}{dt}x^j \circ \gamma(t)\right)\right|}\,dt \, .</math>

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:

:<math>E = \frac{1}{2} \int_a^b \sum_{i,j=1}^ng_{ij}(\gamma(t)) \left(\frac{d}{dt}x^i \circ \gamma(t)\right)\left(\frac{d}{dt}x^j \circ \gamma(t)\right)\,dt \,. </math>

This usage comes from [[physics]], specifically, [[classical mechanics]], where the integral {{mvar|E}} 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 equation]]s may be obtained by applying [[variational principle]]s 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.<ref>{{harvnb|Sternberg|1983}}</ref>