Vector Calculus Operators Grad, Curl, Div in Differential Geometry on Curved Manifolds
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Vector Calculus Operators Grad, Curl, Div in Differential Geometry on Curved Manifolds
In Euclidean space ℝ^3 , vector calculus operators—gradient (grad), curl, and divergence (div)—are defined using partial derivatives. However, on a curved Riemannian or semi-Riemannian manifold, these operators require the use of differential forms, the metric tensor, and the Levi-Civita connection. This formulation generalizes vector ...
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