Covariant and contravariant?

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Can someone explain to me what is the difference between covariant and contravariant vectors ? Thank You
 

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stevendaryl
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Can someone explain to me what is the difference between covariant and contravariant vectors ? Thank You
Given Bill's pointer to a web page talking about the difference, I'm not sure if it's appropriate to add anything, but what I find most useful is not the mathematics for how the two kinds of vectors transform, but what they are good for. The typical use for a regular vector is as a "tangent" or "local approximation" to a parametrized curve--for example, a velocity vector [itex]\vec{v}[/itex] describes how a position as a function of time is behaving locally. The typical use for a covector is a "local approximation" to a scalar field (a scalar field is a real-valued function of location, such as altitude or temperature on the Earth at a given time). In vector calculus in Cartesian coordinates, you would use [itex]\nabla T[/itex] to describe how the scalar field [itex]T[/itex] changes locally. The components of the two types of vectors transform in opposite ways under a change of coordinates.
 

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