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Covariant Derivative & Ricci Identity Differential Geometry Reading Ep. 31

AnthonyMakesVideos 417 lượt xem 2 months ago
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TIMESTAMPS
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Coming up in this video
Intro
Starting to read § 71. Absolute differentiation of tensors of arbitrary order
Covariant derivative of the general mixed tensor of covariant order r and contravariant order s
Gradient of the scalar /phi
§ 72. Further properties of absolute differentiation
Theorem 72.1 The covariant derivatives of the tensors g_{\alpha\beta}, g^{\alpha\beta}, and \dekta_\alpha^\beta are zero tensors.
Proof of Theorem 72.1
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Reacting to viewer comments begins
@magmababy
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@VanDerHaegenTheStampede part 1
part 2
part 3. Arguably my most lucid moment in this video
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Back to reading, now in § 73. Interchange of the order of absolute differentiation. The Ricci identity.
Theorem 73.1 The first covariant derivative of a covariant vector is a symmetric covariant tensor of second order iff that vector is the gradient of a scalar of class r \geq 2.
Proof of Theorem 73.1
Fit check. I call this one "John Foster Dulles".



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