Let $T^\mu_\nu \frac{\partial}{\partial x^\mu} \otimes \mathrm{d}x^\nu$ be a tensor field of type $(1,1)$ on $M$ and let $F:M\to N$ be an diffeomorphism. Show that the induced tensor on $N$ is $$F\ast(T^\mu_\nu \frac{\partial}{\partial x^\mu} \otimes \mathrm{d}x^\nu)=T^\mu_\nu\left(\frac{\partial y^\alpha}{\partial x^\mu}\right)\left(\frac{\partial x^\nu}{ \partial y^\beta} \right)\frac{\partial}{\partial y^\alpha} \otimes \mathrm dy^\beta $$ where ${y^\beta},{x^\alpha}$ are locally coordinates of $N$ and $M$ respectively.
This is a problem in Miko Nakahara's book. My idea is to use the derivation through a function to get the information in the tangent plane. My problem is to express the components of the tensor since my idea is to use the linearity of this.
Push-forwards distribute over products and tensor products. Let $y^\alpha = x^\alpha \circ F^{-1}$ be the coordinates on $N$ adapted to the diffeomorphism $F$ and the initial coordinates $x^\alpha$ on $M$. You have $$F_\ast\left(T^\mu_\nu \frac{\partial}{\partial x^\mu}\otimes{\rm d}x^\nu\right) = F_\ast(T^\mu_\nu) F_\ast\left(\frac{\partial}{\partial x^\mu}\right)\otimes F_\ast({\rm d}x^\nu) = (T^\mu_\nu\circ F^{-1})\left(\frac{\partial y^\alpha}{\partial x^\mu} \frac{\partial}{\partial y^\alpha}\right)\otimes \left( \frac{\partial x^\nu}{\partial y^\beta}\,{\rm d}y^\beta\right),$$which simplifies to $$T^\mu_\nu \frac{\partial y^\alpha}{\partial x^\mu}\frac{\partial x^\nu}{\partial y^\beta}\,\frac{\partial}{\partial y^\alpha}\otimes {\rm d}y^\beta$$as wanted, by abusing the notation and setting $T^\mu_\nu \equiv T^\mu_\nu \circ F^{-1}$, like in the original statement of the problem.