Tensor Calculus Lecture 6c: The Covariant Derivative 2

The paragraph delves into the definition of the covariant derivative, emphasizing its application to tensors of various orders. It explains that for a tensor of order zero, the covariant derivative simplifies to a partial derivative due to the absence of additional Christoffel terms. The speaker reinforces the idea that the covariant derivative of an invariant is equivalent to the partial derivative, highlighting this as property number one. The paragraph also touches on the covariant basis and its utility, and ends with a discussion about the covariant derivative in Cartesian coordinates, where it aligns with the partial derivative when Christoffel symbols vanish.

This section explores the application of the covariant derivative to a third-order tensor and the subsequent contraction, offering two different interpretations. The first interpretation involves applying the covariant derivative first, resulting in additional terms and a fourth-order tensor, followed by contraction. The second interpretation suggests performing the contraction first, simplifying the tensor to an order one before applying the covariant derivative. The paragraph challenges the audience to perform these operations themselves and compare the results, aiming to demonstrate the commutativity of these operations in tensor calculus.

The speaker candidly admits to making a mistake in a tensor calculus operation, emphasizing that learning involves making and correcting errors. The paragraph focuses on the process of operating on lower indices in tensor calculus, particularly when dealing with Christoffel symbols. It discusses the importance of remembering the arrangement and sign changes when expanding terms. The speaker corrects the mistake by re-examining the operation on indices and the application of the covariant derivative, leading to a clearer understanding of tensor operations.

This paragraph investigates the commutativity of covariant derivatives and contractions, a fundamental property in tensor calculus. The speaker presents two interpretations of applying these operations to a tensor and demonstrates that the results are equivalent, regardless of the order in which they are applied. The paragraph reinforces the idea that the covariant derivative and contraction can be interchanged without affecting the outcome, solidifying the concept of commutativity in tensor calculus.

The paragraph discusses additional properties of the covariant derivative, including the metronomic property, which states that the covariant derivative of the metric tensor and related objects results in zero. It also touches on the product rule, also known as the Leibniz rule, and raises a critical question about the commutation of successive covariant derivatives. The speaker sets the stage for a deeper exploration of these properties and their implications in tensor calculus.

This section takes a deep dive into the question of whether covariant derivatives commute when applied successively to a tensor. The speaker outlines the complex expressions that arise from such operations and introduces the Riemann-Christoffel tensor, a fundamental object in differential geometry. The paragraph builds up to a profound conclusion about the nature of these operations and their implications for the study of curvature in space.

The speaker introduces the Riemann-Christoffel tensor and discusses its properties, particularly its transformation behavior across different coordinate systems. The paragraph establishes that the Riemann-Christoffel tensor is indeed a tensor and that it vanishes identically in all coordinate systems where Cartesian coordinates are possible. The discussion leads to the conclusion that the tensor's value is consistent across systems, leveraging the tensor property and the specific behavior of covariant derivatives in Cartesian coordinates.

In this concluding paragraph, the speaker reflects on the implications of the Riemann-Christoffel tensor in the context of curved spaces, as opposed to the Euclidean spaces discussed earlier. The paragraph highlights that in spaces where Cartesian coordinates are not possible, such as in the presence of gravity that warps space, the Riemann-Christoffel tensor does not vanish and becomes a key element in understanding the curvature of the universe. The speaker prepares the audience for further discussions on curved surfaces and the fundamental differences in tensor behavior in such spaces.