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I am trying to understand the notion of a covariant derivative and Christoffel symbols. The proof I am looking at starts out with defining a tensor, T_{mn} = ∂V_{m}/∂x^{n} where V is a covariant vector. I am having a mental block with regard to the indeces. How is it that the derivative of a covariant vector (1st order tensor) be consider to be a 2nd order tensor?