We therefore can give a definition of tensor field, namely as a section of some tensor bundle. Starting with the tangent bundle (the bundle of tangent spaces) the whole apparatus explained at component-free treatment of tensors carries over in a routine way – again independently of coordinates, as mentioned in the introduction. Since the tensor product concept is independent of any choice of basis, taking the tensor product of two vector bundles on M is routine. Given a vector bundle V over M, the corresponding field concept is called a section of the bundle: for m varying over M, a choice of vector For example a vector space of one dimension depending on an angle could look like a Möbius strip as well as a cylinder. There is the idea of vector bundle, which is a natural idea of ' vector space depending on parameters' – the parameters being in a manifold M. The contemporary mathematical expression of the idea of tensor field breaks it down into a two-step concept. It should exist independently of latitude and longitude, or whatever particular 'cartographic projection' we are using to introduce numerical coordinates. 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.
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.
Intuitively, a vector field is best visualized as an 'arrow' attached to each point of a region, with variable length and direction.
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