Transformations in General Theory of Relativity

In our lecture we dicussed that certain elements in GTR transform as certain other things, i.e.

• Christoffel symbol transforms as an affine connection
• covariant derivative of a contravariant vector transforms as a tensor
• covariant derivative of a covariant vector transforms as a tensor
• equation of motion for a particle in a grav. field transforms as a contravariant vector

I only understand the last of those examples. Consider the equation of motion
begin{equation}
Phi equiv frac{d^2 x^mu}{dtau^2} + Gamma^mu_{nulambda}frac{dx^nu}{dtau}frac{dx^lambda}{dtau} = 0
end{equation}

I can show that the transformation yields
begin{equation}
frac{d^2 x’^mu}{dtau^2} + Gamma’^mu_{nulambda}frac{dx’^nu}{dtau}frac{dx’^lambda}{dtau}
= …
= frac{dx’^mu}{dx^alpha}left( frac{d^2x^alpha}{dtau^2} + Gamma^alpha_{betagamma} frac{dx^beta}{dtau} frac{dx^gamma}{dtau} right).
end{equation}
This is equal to:
begin{equation}
Phi’^mu = frac{dx’^mu}{dx^alpha} Phi^alpha.
end{equation}

Since a contravariant vector itself transforms as
begin{equation}
V’^mu = frac{dx’^mu}{dx^alpha}V^alpha
end{equation}
I can easily see the pattern here. I do however fail to understand the other three examples.