The objective is to derive $$p_k=\frac{\partial\phi}{\partial q_k},\quad Q_k=\frac{\partial\phi}{\partial P_k},\quad \tilde{H}=H+\frac{\partial\phi}{\partial t},$$ where $\phi=\phi(t,q,P)$. I successfully derived $p_k$ as follows. Firstly $$\frac{d\phi}{dt}=\sum_{k=1}^{n\in\mathbb{N}}\left(\frac{\partial\phi}{\partial q_k}\dot{q}_k+\frac{\partial\phi}{\partial P_k}\dot{P}_k\right)+\frac{\partial\phi}{\partial t}.$$ Secondly $$\frac{d\phi}{dt}=\sum_{k=1}^{n\in\mathbb{N}}(p_k\dot{q}_k-P_k\dot{Q}_k)+H(t,q,p)-\tilde{H}(t,Q,P).$$ Hence $$p_k\dot{q}_k=\frac{\partial\phi}{\partial q_k}\dot{q}_k,\quad P_k\dot{Q}_k=-\frac{\partial\phi}{\partial P_k}\dot{P}_k,\quad\text{and}\quad\frac{\partial\phi}{\partial t}=H(t,q,p)-\tilde{H}(t,Q,P).$$ Thus $$p_k=\frac{\partial\phi}{\partial q_k},$$ as desired, but $$P_k\dot{Q}_k=-\frac{\partial\phi}{\partial P_k}\dot{P}_k\not\Leftrightarrow Q_k=\frac{\partial\phi}{\partial P_k}.$$ Where did I go wrong? I'm following the discussion on symplectic maps in section 8.3 of Bruce van Brunt's book on Calculus of Variations.
Here, $q$ and $p$ are the coordinates of the Hamiltonian $H(t,q,p)$ before transformation, and $Q$ and $P$ are the coordinates of the transformed Hamiltonian $\tilde{H}(t,Q,P)$.