- Line element
A line element in
mathematics can most generally be thought of as the square of the change in a position vector in anaffine space equated to the square of the change of thearc length . An easy way of visualizing this relationship is by parametrizing the givencurve byFrenet's formulas . As such, a "line element" is then naturally a function of the metric, and can be related to thecurvature tensor .The most well known line elements are those of cartesian planar and spatial coordinates. They are given by
planar:ds^2= dx^2 +dy^2
spatial:ds^2= dx^2 +dy^2 +dz^2
Other line elements are given by:
flat polar: ds^2= dr^2 +r^2 d heta ^2
spherical polar: ds^2=dr^2+r^2 d heta ^2+ r^2 sin^2 heta d phi ^2
cylindrical polar:ds^2=dr^2+ r^2 d heta ^2 +dz^2
The most general 2- dimensional (coordinates (χ,ψ)) metric is given by
ds^2= f ( chi , psi )d chi ^2 + g ( chi , psi )d chi d psi + h ( chi , psi ) d psi ^2
Line elements in physics
Line elements are used in physics, especially in theories of gravitation such as
general relativity , wherespacetime is modelled as a curved manifold with a metric.ee also
*
First fundamental form
*Metric tensor References
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