Given a contravariant basis , its dual covariant basis is given by
where is the metric and is the mixed Kronecker delta. In Euclidean space with an orthonormal basis,
so the basis and its dual are the same.
Given a contravariant basis , its dual covariant basis is given by
where is the metric and is the mixed Kronecker delta. In Euclidean space with an orthonormal basis,
so the basis and its dual are the same.
Weisstein, Eric W. "Dual Basis." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DualBasis.html