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What is the general difference or importance between using christoffel symbols of the first kind and those of the second kind in terms of geometry and their application. The christoffel symbols of the second are identical to those of the first except with the inverse metric tensor in front, correct?Why have the both the first and second kind? what are the first used for and the second used for in general? I know the importance of the second kind in terms of the covariant derivative but what use does the first kind have in physics and math.