Are you an EPFL student looking for a semester project?
Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.
Concept
Covariance and contravariance of vectors
Summary
In physics, especially in multilinear algebra and tensor analysis, covariance and contravariance describe how the quantitative description of certain geometric or physical entities changes with a change of basis. In modern mathematical notation, the role is sometimes swapped.
A simple illustrative case is that of a vector. For a vector, once a set of basis vectors has been defined, then the components of that vector will always vary opposite to that of the basis vectors. A vector is therefore a contravariant tensor. Take a standard position vector for example. By changing the scale of the reference axes from meters to centimeters (that is, dividing the scale of the reference axes by 100, so that the basis vectors now are meters long), the components of the measured position vector are multiplied by 100. A vector's components change scale inversely to changes in scale to the reference axes, and consequently a vector is called a contravariant tensor.
In contrast, a covector, also called a dual vector, has components that vary with the basis vectors in the corresponding vector space. It is an example of a covariant tensor. A covector is an object that represents a linear map from vectors to scalars. It is actually not a vector, but an object that lives in a dual vector space. Some good examples of covectors are dot product operators involving vectors. For example if is a vector, then a corresponding object in the dual space would be the linear operator . Sometimes, the components of the covector are referred to as the covariant components of , although this is potentially misleading, (due to a vector having components that always vary in the contravariant sense). Despite potential confusion, this is what will be meant when the "covariant components of a vector" are referred to herein.
The gradient is often cited as an example of a covector, but this is incorrect.
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.