If I have a tensor field $g$ of type $(0,2)$, in local coordinates expressed as $g=g_{ij}dx^i\otimes dx^j$, changing coordinates ($x^i\rightarrow y^j$) I obtain the following: $$ g=g_{ij}(\frac{\partial x^i}{\partial y^k}dy^k)\otimes(\frac{\partial x^j}{\partial y^h}dy^h)$$ $\textbf{My questions:}$ Sorry in advance if they are stupid questions but I am new in studying differential geometry...
can I consider that the tensor product above is $\textbf{bilinear}$ and so can I write the following? $$g=g_{ij}(\frac{\partial x^i}{\partial y^k}dy^k)\otimes(\frac{\partial x^j}{\partial y^h}dy^h)=g_{ij}\frac{\partial x^i}{\partial y^k}\frac{\partial x^j}{\partial y^h}dy^k\otimes dy^h$$
let $\tilde g_{kh}=\frac{\partial x^i}{\partial y^k}\frac{\partial x^j}{\partial y^h}$ I have tried to prove that it is equal to $\tilde g_{hk}$ in this way: I need $$\frac{\partial x^i}{\partial y^k}\frac{\partial x^j}{\partial y^h}=\frac{\partial x^i}{\partial y^h}\frac{\partial x^j}{\partial y^k}$$ I have done the following: $$\frac{\partial x^j}{\partial y^h}=\color{red}{\frac{\partial y^k}{\partial x^i}}\frac{\partial x^i}{\partial y^k}\frac{\partial x^j}{\partial y^h}=\color{red}{\frac{\partial y^k}{\partial x^i}}\frac{\partial x^i}{\partial y^h}\frac{\partial x^j}{\partial y^k}\iff \frac{\partial x^j}{\partial y^h}=\delta^k_h \frac{\partial x^j}{\partial y^k}=\frac{\partial x^j}{\partial y^h}$$ but in the second equalities I am supposing what really I have to prove...so I am not convinced of what I have done
Please can you help me in solving these two questions?