🎬how the metric tensor allows for raising and lowering indices by seeing two distinct calculations

The metric tensor is the fundamental tool for performing consistent calculations in a curved space. It ensures that basic mathematical operations like the dot product and the covariant derivative remain physically consistent regardless of how you represent the vectors. The demo highlights this by showing that even though you can use two different mathematical formalisms (contravariant vs. covariant vectors) to calculate the same physical quantity (the rate of change of a scalar field), the results are identical. This is because the metric tensor provides a consistent way to "raise and lower indices," which is the formal way of saying it allows you to move between these different vector representations without changing the underlying physics.

The Metric Tensor Covariant Derivatives and Tensor Densities | Condensed Notesviadean on Notion