dsaun777 said:Can you take any non invariant quantity like components and take the covariant derivative of them and arrive at an invariant tensor quantity?
Sorry my brain is not working long day...PeterDonis said:You can't take covariant derivatives of components. Covariant derivatives operate on tensors.
A covariant derivative is a mathematical operation that is used to measure how a tensor field changes as one moves along a specified direction in a curved space. It takes into account the curvature of the space and adjusts for it, making it a more accurate measure than a regular derivative.
Limits on making a tensor refer to the restrictions on the types of operations that can be performed on a tensor field. These limits are necessary to maintain the tensor's properties, such as its rank and transformation behavior, and to ensure that it remains a valid representation of the underlying physical system.
The covariant derivative takes into account the limits on making a tensor by incorporating a correction term that accounts for the curvature of the space. This correction term ensures that the resulting tensor field is still a valid representation of the physical system and maintains its properties.
A regular derivative only takes into account the change in a tensor field along a particular direction, while a covariant derivative also takes into account the curvature of the space. This makes the covariant derivative a more accurate measure of change in a curved space.
The covariant derivative is important in physics because it allows us to accurately describe and analyze physical systems in curved spaces, such as general relativity. It also ensures that the equations and laws of physics remain consistent and valid in these curved spaces.