ABSTRACT
Think about a collection of points in a space. Usually the ordinary points of our three-dimensional space are what we shall need, but sometimes the “points” might be the states of a quantum mechanical Hilbert space. (You might have to deal with a manifold, hyperspace, variety, etc.) Give the points labels so that you can find them again. Say we use xi where i takes values in the appropriate range, for example, i = 1, 2, 3. Then move the points of the space (I take the active viewpoint), and the point that had labels xi now has xi ′ as labels. For the physics we have in mind, we assume that the functions give the new coordinates xi ′ in terms of the old xi . https://www.w3.org/1998/Math/MathML"> x ′ r = f r ( x 1 , x 2 , ... , x N ) ( r = 1 , ... , N ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429184550/c5a9cf84-23e1-4064-9971-05f6815a4e54/content/math3_1_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> are continuous and differentiable (real, if the coordinates are real), and the functional determinant (Jacobian) https://www.w3.org/1998/Math/MathML"> | ∂ x ″ ∂ x ′ ⋯ ∂ x ″ ∂ x N ⋮ ⋱ ⋮ ∂ x ′ N ∂ x ′ ⋯ ∂ x ′ N ∂ x N | https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429184550/c5a9cf84-23e1-4064-9971-05f6815a4e54/content/math3_2_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> never vanishes. (Of course, you put up with singularities such as that at the origin in the change between Cartesian and spherical polar coordinates.) In principle you can then solve to get https://www.w3.org/1998/Math/MathML"> x r = g r ( x ″ , x ′ 2 , ... , x ′ N ) ( r = 1 , 2 , ... , N ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429184550/c5a9cf84-23e1-4064-9971-05f6815a4e54/content/math3_3_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> and this is called the implicit function theorem. Now what are the objects that are of interest to a physicist? Those things that have simple properties under the transformations. We will list some of the more important ones.
