dg.differential geometry - Change of coordinates introduced through dx

Ok here are the partial derivatives $k_M$:

they are dual to ${b^M} = {dy^mu,dtheta^alpha}$ ie:

$b^M(k_N)=delta_N^M$

Writing $k_N=(k_N)^{mu}frac{partial}{partial x^mu}+(k_N)^{alpha}frac{partial}{partial theta^alpha}$ and solving $b^nu(k_mu)=delta^nu_mu,;b^mu(k_alpha)=0; ; b^alpha(k_nu)=0;;b^alpha(k_beta)=delta^alpha_beta$ one finds:

$(k_mu)^nu=delta^nu_mu,;(k_mu)^alpha=0;;(k_alpha)^beta=delta^beta_alpha;;(k_alpha)^nu=-eta_alpha^{;;mu}$

So:

$frac{partial}{partial y^mu} = frac{partial}{partial x^mu}$

$frac{partial}{partial y^alpha} = -eta^{;;mu}_alpha frac{partial}{partial x^mu} + frac{partial}{partial theta^alpha}$