Video 38 - Metrinilic Property - Part 2

In this segment, the video discusses the covariant derivative of the Kronecker Delta function, which is shown to be zero due to the metric property. The covariant derivative is applied to the product of basis vectors, and it's demonstrated that the result is zero because the covariant derivative of each basis vector is zero. This property is then used to simplify expressions involving the covariant derivative of products involving the Kronecker Delta. Practical applications are also explored, such as simplifying expressions with the covariant derivative of a product of Delta and a tensor, where the Delta factor can be moved outside the derivative due to its metric property.

This paragraph delves into the practical applications of the metric property in tensor calculus, specifically how it simplifies the process of raising and lowering indices. The video provides examples of how the covariant derivative can eliminate symmetric tensor factors, allowing for the direct manipulation of indices without intermediate steps. The contravariant metric tensor is also discussed, showing that it possesses the same metric property as the covariant metric tensor, and this property is used to simplify expressions involving the covariant derivative of metric tensors.

The video introduces a new operation called the contravariant derivative, which allows for the formation of contractions using covariant vector components. This is contrasted with the covariant derivative, which requires a contravariant vector component. The contravariant derivative is defined in terms of the covariant derivative and the metric tensor, and it is shown to be equivalent to the covariant derivative when contracted with a covariant vector component. The concept of divergence is also extended to include this new operation, allowing for the flipping of indices between the derivative operation and the operand.

This section discusses the relationship between the Kronecker Delta, metric tensors, and basis vectors. It explains how the Kronecker Delta can be related to the metric tensors by considering it as a tensor and performing index raising and lowering operations. The video demonstrates that the Kronecker Delta is identical to a mixed metric tensor, which has both contravariant and covariant indices. This connection reinforces the idea that the Kronecker Delta is part of the metric family of tensors and highlights the fundamental role of metric tensors in tensor calculus.

The video concludes by summarizing the metronic property and its implications for tensor operations. It emphasizes that the covariant derivative of the Kronecker Delta and both the covariant and contravariant metric tensors is zero, allowing these tensors to be treated as constants within the covariant derivative operation. This property simplifies the process of flipping indices and performing tensor operations. The video also introduces the concept of the mixed metric tensor and shows its equivalence to the Kronecker Delta, further illustrating the interconnectedness of these tensor calculus objects.

In the final paragraph, the video provides a recap of the main points covered regarding the metronic property. It reiterates how the covariant derivative of the Kronecker Delta and metric tensors is zero, and how this property allows for the simplification of tensor expressions. The contravariant derivative is also discussed as a means to extend the definition of divergence and to work with covariant objects more conveniently. The video ends with a clear explanation of why the Kronecker Delta is considered part of the metronic family, tying it back to the fundamental role of metric tensors in tensor calculus.