Covariant Differentiation in Differential Calculus

In mathematics, one way in which we talk about constancy is we say that something has derivative equal to zero. But there are multiple different kinds of derivatives, and those are called coveriantDerivatives. And the covarient derivatives can vary along with the function - so if one function was constant under one derivative, a new function would be constant under the new derivative. That would be called gauge in variance, or gauge covarient. O.K., i think wh you've just defined is a concept called parallel translation. I'm going to throw out a general definition, and maybe we going o start a riff on it.

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