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.
The general idea of tensor field combines the requirement of richer geometry — for example, an ellipsoid varying from point to point, in the case of a metric tensor — with the idea that we don't want our notion to depend on the particular method of mapping the surface. It should exist independently of latitude and longitude, or whatever particular 'cartographic projection' we are using to introduce numerical co-ordinates.
Read more about this topic: Tensor Field
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