Tensor Analysis Lecture 6 : Tensor calculus | Parallel transport | Covariant differentiation скачать в хорошем качестве

Tensor Analysis Lecture 6 : Tensor calculus | Parallel transport | Covariant differentiation

Covariant differentiation and parallel transport are fundamental tensor calculus tools used to analyze tensors in curved spaces, where standard derivatives fail because vectors at different points belong to different tangent spaces. Covariant differentiation adds correction terms (Christoffel symbols) to partial derivatives, measuring the intrinsic change of a tensor field. Unlike partial derivatives, the covariant derivative ensures that the result is a tensor, accounting for the change in basis vectors due to coordinate curvature. It acts as a derivative that obeys product rules and reduces to a partial derivative in flat Cartesian coordinates. Parallel transport moves a vector along a curve while keeping it "parallel" to itself, meaning its covariant derivative along the curve is zero. Parallel transport preserves the inner product (length) of vectors and angles between them.