Coordinates can depend directly on dimensions {contravariant, tensor}. Coefficients a times basis vectors e result in coordinates x: a(i) * e(i) = x(i).
covariant
Covariant components relate to contravariant components. If basis vectors are orthogonal, covariant components and contravariant components are equal: a(i) = x(i) / e(i).
If basis vectors are curved coordinates, then a(i) = g(i,j) * a(j), where g(i,j) depend on basis vectors e(i) and e(j). Some g(i,j) components are for covariance, some for contravariance, and some for both. g(i,j) tensor relates basis vectors. g(i,j) elements are functions of curved-space positions.
Euclidean space
g(i,j) elements are 1 or 0 for flat space with orthogonal basis vectors.
Mathematical Sciences>Calculus>Vector>Tensor>Operations
3-Calculus-Vector-Tensor-Operations
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Date Modified: 2022.0224