Editing Tensor field (section)

== Geometric introduction ==

Intuitively, a vector field is best visualized as an "arrow" attached to each point of a region, with variable length and direction. One example of a vector field on a [[curved space]] is a weather map showing horizontal wind velocity at each point of the Earth's surface.

Now consider more complicated fields. For example, if the manifold is Riemannian, then it has a metric field <math>g</math>, such that given any two vectors <math>v, w</math> at point <math>x</math>, their inner product is <math>g_x(v, w)</math>. The field <math>g</math> could be given in matrix form, but it depends on a choice of coordinates. It could instead be given as an ellipsoid of radius 1 at each point, which is coordinate-free. Applied to the Earth's surface, this is [[Tissot's indicatrix]].

In general, we want to specify tensor fields in a coordinate-independent way: It should exist independently of latitude and longitude, or whatever particular "cartographic projection" we are using to introduce numerical coordinates.