Proof that Xp|k is a type (1, 1) tensor

We already know from the notes that DXp/dt is a type (1, 0) tensor, so we write down its transformation rule:

Using the chain rule, we can rewrite this as

Now, this is true for any path whatsoever, entitling us to cancel the terms da/dt. If you cancel these terms, and stare at what is left, you will see

which is the transofrmation fule for a type (1, 1) tensor!


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