🧪Tensors in Cartesian Coordinates and Their Integration

In Cartesian coordinates, tensor analysis is simpler because basis vectors are constant and orthonormal. This eliminates the distinction between covariant and contravariant indices. Tensor transformations are also straightforward, using constant coefficients instead of partial derivatives. For tensor integration, components can be integrated individually, unlike in general coordinates where basis vectors vary with position. The volume element is a simple product of differentials in Cartesian coordinates, but it includes the metric determinant's square root in general coordinates to remain a scalar. The text concludes by noting a key limitation of tensor integration in general spaces: the result doesn't belong to a single point and can't be expressed in a single basis, a problem that scalar integrals don't have.

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Tensors in Cartesian Coordinates and Their Integration | Condensed Notesviadean on Notion