basis vector

Space has dimensions. Vectors {basis vector} can lie along coordinate axes. Other vectors are basis-vector linear combinations.

contravariant

For point, coefficients a(i) times basis vectors e(i) result in point coordinates x(i): a(i) * e(i) = x(i), where i is number of dimensions. Points can move. New coefficients x(i) of same dimensions e(i) relate to old coefficients a(i): a(i) * e(i) = x(i).

covariant

For points, coordinate system can change. New-dimension coefficients a(i) relate to old dimensions e(i), because new dimensions are linear transformations x(i) of old dimensions: a(i) = x(i) * e(i).

relation

Covariant components relate to contravariant components. If basis vectors are orthogonal, covariant components and contravariant components are equal. If basis vectors are curved coordinates, a(i) = g(i,j) * a(j), where a(i) and a(j) are coefficients and g(i,j) is tensor relating basis vectors e(i) ... e(j). Some g(i,j) components are for covariance, some for contravariance, and some for both. g(i,j) elements are functions of curved-space positions. g(i,j) elements are 1 or 0 for flat space with orthogonal basis vectors.

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