Starting with holomorphic function in complex-plane region, the function can extend to other domains by moving along path to points that allow overlapping regions {analytic continuation}. Different paths result in different extensions.
Tensors can project onto coordinate systems, with basis vectors, to find coefficients {contravariance}|. Spatial-coordinate differentials dx^i make simplest contravariant vector {rank one tensor}. Contravariant coefficients contracted with metric tensor give covariant coefficients.
Transforming functions can result in same product {covariance}| times a constant or a power of determinant {modulus, covariance}. For tensors, covariance transforms basis-vector coefficients into other basis-vector coefficients.
Function groups {family of functions} can differ by algebraic parameter.
Transforming functions can have same results {invariance, function}. Energy conservation requires that total energy be invariant. Functions can remain the same under linear transformations {algebraic invariant}. Invariants are covariants of order zero.
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Date Modified: 2022.0225