The Notion of a Riemannian Metric
In the preceding chapter we associated with each curvilinear coordinate system x = (x 1, ... , x n) on a domain C a smooth functional matrix G(x), and these matrices, at each point, behave as a quadratic form under a change-of-coordinate transformations. The matrices G(x) arose from the formula for the element of length of a curve, and we saw that the length of a curve can be calculated in a curvilinear coordinate system using the G(x). The property of transforming at each point like a quadratic form turns out to be of major importance.
KeywordsQuadratic Form Bilinear Form Tangent Vector Parametrized Surface Euclidean Plane
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