# Expression of $n$-form with two charts.

Let $$(U , \varphi)$$ and $$(V , \psi)$$ be two charts on a $$n$$-dimensional differentiable manifold $$M$$, with $$U \cap V \neq \emptyset$$, such that $$\varphi = (x_1 , \ldots , x_n)$$ and $$\psi = (y_1 , \ldots , y_n)$$. We have two elements in $${\Lambda}^n(T_pM)$$ ($$p \in U \cap V$$): $${(d x_1)}_p \wedge \ldots \wedge {(d x_n)}_p \qquad \mbox{ and } \qquad {(d y_1)}_p \wedge \ldots \wedge {(d y_n)}_p\mbox{.}$$ How can I show that $${(d y_1)}_p \wedge \ldots \wedge {(d y_n)}_p = \left(\det d {(\psi \circ {\varphi}^{- 1})}_{\varphi(p)}\right) {(d x_1)}_p \wedge \ldots \wedge {(d x_n)}_p?$$ I have got the equality $${(d y_1)}_p \wedge \ldots \wedge {(d y_n)}_p = \lambda(p) {(d x_1)}_p \wedge \ldots \wedge {(d x_n)}_p\mbox{,}$$ where $$\lambda(p) = \det {\left({(d y_i)}_p {\left(\frac{\partial}{\partial x_j}\right)}_p\right)}_{i , j = 1}^n\mbox{.}$$ I am using the notation $${\left\{{\left(\frac{\partial}{\partial x_i}\right)}_p\right\}}_{i = 1}^n$$ basis for $$T_pM$$ (using the chart $$(U , \varphi)$$) and $${\{{(d y_i)}_p\}}_{i = 1}^n$$ dual basis of $${\left\{{\left(\frac{\partial}{\partial y_i}\right)}_p\right\}}_{i = 1}^n$$ for $$T_p^*M$$ ($$= {(T_pM)}^*$$).

How can I show that $$\lambda(p) = \det d {(\psi \circ {\varphi}^{- 1})}_{\varphi(p)}$$?