# Covariant and contravariant?

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

bhobba
Mentor
stevendaryl
Staff Emeritus
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 $\vec{v}$ 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 $\nabla T$ to describe how the scalar field $T$ changes locally. The components of the two types of vectors transform in opposite ways under a change of coordinates.