Calculating the "curvature" of a coordinate system to define derivatives (covariant differentiation).
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$$\nabla_j V^i = \frac\partial V^i\partial x^j + \Gamma^i_jk V^k$$ where $\Gamma^i_jk$ are the Christoffel symbols. In flat space (Cartesian coordinates), the Christoffel symbols vanish, so $\nabla_j V^i = \partial_j V^i$.
Introduction to Tensor Analysis and the Calculus of Moving Surfaces