Covariance and Contravariance of Vectors - Covariant and Contravariant Components of A Vector With A Metric

Covariant and Contravariant Components of A Vector With A Metric

In a vector space V over a field K with a bilinear form g : V × VK (which may be referred to as the metric tensor), there is little distinction between covariant and contravariant vectors, because the bilinear form allows covectors to be identified with vectors. That is, a vector v uniquely determines a covector α via

for all vectors w. Conversely, each covector α determines a unique vector v by this equation. Because of this identification of vectors with covectors, one may speak of the covariant components or contravariant components of a vector, that is, they are just representations of the same vector using reciprocal bases.

Given a basis f = (X1,...,Xn) of V, there is a unique reciprocal basis f# = (Y1,...,Yn) of V determined by requiring that

the Kronecker delta. In terms of these bases, any vector v can be written in two ways:

\begin{align}
v &= \sum_i v^iX_i = \mathbf{f}\,\mathbf{v}\\
&=\sum_i v_iY^i = \mathbf{f}^\sharp\mathbf{v}^\sharp.
\end{align}

The components vi are the contravariant components of the vector v in the basis f, and the components vi are the covariant components of v in the basis f. The terminology is justified because under a change of basis,


Read more about this topic:  Covariance And Contravariance Of Vectors

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