Quick, simple question about contraviant and covariant components

In summary, the covariant and contravariant components of a vector can be seen as the vector multiplied by a matrix of linearly independent vectors that span the vector space and its inverse, respectively. This is why they are equal when the basis is the normalized mutually orthogonal one, as the matrix is the identity matrix. The transformation between the tangent and cotangent spaces is performed by the metric and its inverse, with the zero'th component of Vμ being determined by the metric.
  • #1
enfield
21
0
Can the covariant components of a vector, v, be thought of as v multiplied by a matrix of linearly independent vectors that span the vector space, and the contravariant components of the same vector, v, the vector v multiplied by the *inverse* of that same matrix?

thinking about it like that makes it easy to see why the covariant and contravariant components are equal when the basis is the normalized mutually orthogonal one, for example, because then the matrix is just the identity one, which is its own inverse.

that's what the definitions i read seem to imply.

Thanks!
 
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  • #2
The contravariant and covariant components of a vector are linear combinations of each other, but the transformation is performed by the metric and the inverse metric. It is a change of basis between the tangent and cotangent spaces.

The zero'th component of Vμ is given by

V0= g0aVa = g00V0+g01V1+g02V2+g03V3

If the metric is the identity matrix then the components are the same.
 
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What are contravariant and covariant components?

Contravariant and covariant components are terms used in tensor analysis to describe how the components of a tensor change under a coordinate transformation. In simple terms, contravariant components change in the opposite direction as the coordinates, while covariant components change in the same direction as the coordinates.

How do contravariant and covariant components differ?

Contravariant and covariant components differ in their transformation rules under a coordinate transformation. Contravariant components have a negative power of the Jacobian matrix in their transformation rule, while covariant components have a positive power.

What are some examples of contravariant and covariant components?

An example of a contravariant component is the velocity of a particle, which changes in the opposite direction as the coordinates. An example of a covariant component is the gradient of a scalar function, which changes in the same direction as the coordinates.

Why are contravariant and covariant components important?

Contravariant and covariant components are important in tensor analysis because they allow for the calculation of tensors in different coordinate systems. They also help in understanding the geometric properties of tensors and their transformations.

How can I convert between contravariant and covariant components?

To convert between contravariant and covariant components, you can use the metric tensor, which relates the two sets of components. By contracting the metric tensor with the components, you can transform them from one set to the other.

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