Proof that X^p_|k is a type (1, 1) tensor

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



dj dt

+

j r i

r

di dt

=

j xp

dXp dt

+

p h k

Xh

dxk dt

Using the chain rule, we can rewrite this as



j a

+

j r i

r

i a

da dt

=

j xp

Xp xb

+

p h k

Xh

xk xb

xb a

da dt

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



j|a

=

Xp|b

j xp

xb a

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