Recall that the tension field of a function $f:(M,g)\rightarrow (N,G)$ is given in local coordinates by \begin{align*} & \Delta_gf^k+g^{ij}\hat{\Gamma}^k_{mn}\partial_if ^m\partial_jf^n \\ \end{align*} Where $\hat{\Gamma}$ denote the Christoffel symbol associated to $G$, and $\partial_i$ denotes the partial derivative with respect to $x^i$, coordinate of the domain.
I want to show that this is a vector field on $N$ by showing that it transforms as one, under change of coordinates in the target space.
After a lenghty calculation (see Calculation) , denoting by $y$ the starting local coordinates, and by $\phi$ the change of coordinates map, I obtained: \begin{align*} \tilde{\tau}(f)^t & =\Delta_g(\phi\circ f)^t+g^{ij}\tilde{\Gamma}^t_{mn}\partial_i(\phi \circ f)^m \partial_j(\phi \circ f)^n \\ & = \frac{\partial\phi^t}{\partial y^k}\Big(\Delta_gf^k+g^{ij}\hat{\Gamma}^k_{mn}\partial_if ^m\partial_jf^n \\ &\quad \qquad\qquad+\color{red}{\frac{\partial f^k}{\partial\phi^t}g^{ij}\frac{\partial^2\phi^t}{\partial y^r\partial y^l}\cdot \thinspace \partial_if^r \partial_jf^l + g^{ij}\frac{\partial^2f^k}{\partial\phi^l\partial\phi^r} \partial_i(\phi \circ f)^r \partial_j(\phi \circ f)^l}\Big). \end{align*} If I am not mistaken, the first two terms are the ones which should appear, wheras the ones in red should vanish, but I have no clue how to get rid of them, as I have tried everything I had in my toolbox, but didn't get anywhere.
Can someone help me out?
PS: I know there exists a intristic definition of $\tau(f)$ as $tr_g\nabla d(f)$, but I want to go the explicit way.