Difference Between Covariant & Contravariant Vectors Explained

In summary, covariant and contravariant vectors have different uses and transform in opposite ways under a change of coordinates. Covariant vectors are used to approximate scalar fields while contravariant vectors are used to approximate parametrized curves.
  • #1
LeonPierreX
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Can someone explain to me what is the difference between covariant and contravariant vectors ? Thank You
 
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  • #3
LeonPierreX said:
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.
 

1. What is the difference between covariant and contravariant vectors?

Covariant and contravariant vectors are two types of vectors that describe how a vector changes when the coordinate system is transformed. The main difference between them lies in how they transform under a change of coordinates.

2. How do covariant and contravariant vectors transform under a change of coordinates?

Covariant vectors transform with the coordinate system, meaning they change in the same way as the coordinates do. Contravariant vectors, on the other hand, transform against the coordinate system, meaning they change in the opposite way as the coordinates do.

3. Can you give an example of a covariant and contravariant vector?

An example of a covariant vector is the position vector, which transforms in the same way as the coordinates under a change of coordinates. An example of a contravariant vector is the gradient of a scalar field, which transforms in the opposite way as the coordinates.

4. How are covariant and contravariant vectors related?

Covariant and contravariant vectors are related through the metric tensor, which describes the relationship between the two types of vectors. The metric tensor is used to convert between covariant and contravariant components of a vector.

5. Why is it important to understand the difference between covariant and contravariant vectors?

Understanding the difference between covariant and contravariant vectors is important in many areas of physics and mathematics, such as tensor calculus and general relativity. It allows for a deeper understanding of how vectors behave under different coordinate systems and is essential for formulating and solving problems in these fields.

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