On the page covering the topic on Covariance and contravariance of vectors,
/info/en/?search=Covariance_and_contravariance_of_vectors, it has been said contravariance vectors have components that "transform as the coordinates do" and inversely to the transformation of the reference axes. What are the difference between (or the definition of) components of the vectors, coordinates of the vectors and reference axes of the vectors?
Please help me with...
As I understand, when the basis vectors (sometimes called "reference axes") are transformed, the coordinates (of everything else) change inversely. So a position vector transforms inversely as the basis vectors (its components are thus contravariant), and the same as the coordinates (which may seem like a tautology). A gradient vector transforms the same as the basis vectors (its components are thus covariant), and inversely as the coordinates.
The best definition of coordinates and components I have seen is the following:
If , the scalar is called the i-th coordinate of , and is called the i-th component of . Generally, coordinates and components depend on the choice of the entire basis and cannot be determined from individual vectors in the basis. Because of the rather simple correspondence between coordinates and components there is a tendency to confuse them and to use both terms for both concepts. Since the intended meaning is usually clear from context, this is seldom a source of difficulty.[1]
On the page covering the topic on Covariance and contravariance of vectors,
/info/en/?search=Covariance_and_contravariance_of_vectors, it has been said contravariance vectors have components that "transform as the coordinates do" and inversely to the transformation of the reference axes. What are the difference between (or the definition of) components of the vectors, coordinates of the vectors and reference axes of the vectors?
Please help me with...
As I understand, when the basis vectors (sometimes called "reference axes") are transformed, the coordinates (of everything else) change inversely. So a position vector transforms inversely as the basis vectors (its components are thus contravariant), and the same as the coordinates (which may seem like a tautology). A gradient vector transforms the same as the basis vectors (its components are thus covariant), and inversely as the coordinates.
The best definition of coordinates and components I have seen is the following:
If , the scalar is called the i-th coordinate of , and is called the i-th component of . Generally, coordinates and components depend on the choice of the entire basis and cannot be determined from individual vectors in the basis. Because of the rather simple correspondence between coordinates and components there is a tendency to confuse them and to use both terms for both concepts. Since the intended meaning is usually clear from context, this is seldom a source of difficulty.[1]