Physical quantities or coordinates can transform from one coordinate system to another {coordinate transformation}. First coordinate-system vector components are linear functions of second coordinate-system components.
tensor
Tensor coefficients are weights by which to multiply old variables to get new variables. Tensor-term number is old-component number times new-component number. Scalar product of outer-product tensor and old basis vectors obtains new basis-vector scalars.
projection
Linear transformation projects old onto new. Linear transformation is affine geometry.
contravariant
Contravariant component {contravariant}, such as dx, multiplies with tensor. Terms with same index, such as ii, have coefficient one. Terms with different indexes, such as ij, have coefficient zero. In contravariant transformation, only diagonal terms remain. Contravariant component sum is vector expressed in old basis vectors. Diagonal terms are scalars for new basis vectors.
covariant
Covariant component {covariant}, such as partial derivative, is contravariant component times tensor. Terms with different indexes, such as ij, have coefficient one. Terms with same index, such as ii, have coefficient zero. In covariant transformation, diagonal terms are not present. Covariant component sum is weight matrix. Non-diagonal terms are weights.
covariance and contravariance
Covariant means that different old and new components interact. Contravariant means that same old and new components interact. Together, they account for all interactions. Contravariant reduces dimension by one. Covariant does not change dimensions. If components are orthogonal, as in Euclidean space, covariant and contravariant components are the same. Contravariant or covariant transformation does not change symmetrical-tensor value. Contravariant or covariant transformation only changes sign of odd number of skew-symmetrical tensor transformations.
Mathematical Sciences>Calculus>Vector>Tensor>Operations
3-Calculus-Vector-Tensor-Operations
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Date Modified: 2022.0224