# Equivalence of definitions of oriented atlas.

In the book An introduction to Manifolds by Loring Tu an oriented atlas is defined as follows.

An atlas is oriented if for any two overlapping charts $$(U,\phi =(x^1,\ldots,x^n))$$ and $$(V,\psi=(y^1,\ldots,y^n))$$ the Jacobian determinant $$\operatorname{det}[\partial y^i/\partial x^j]$$ is strictly positive on $$U\cap V$$

In my course notes this is defined as

An atlas is oriented if for two overlapping charts as above $$\operatorname{det}D(\psi\circ\phi^{-1})>0$$ on $$\phi(U\cap V)$$.

I am struggling to show this is equivalent. It seems like a straightforward calculation, but I am not getting there. Calculating $$D(\psi\circ\phi^{-1})$$ I get $$\begin{pmatrix}\frac{\partial \phi^{-1}}{\partial x_1}&\dots&\frac{\partial \phi^{-1}}{\partial x_1}\\\vdots &\ddots & \vdots\\\frac{\partial \phi^{-1}}{\partial x_n}&\dots& \frac{\partial \phi^{-1}}{\partial x_n}\end{pmatrix}$$ Is this correct or am I doing something wrong? How can I show that this determinant has the same sign as the one from the first definition?

A chart is a homeomorphism $$\phi : U \to U'$$, where $$U \subset M$$ is open and $$U' \subset \mathbb R^n$$ is open. Given another chart $$\psi : V \to V'$$, we get the transition map $$\psi \circ \phi^{-1} : \phi(U \cap V) \to \psi(U \cap V) .$$ It has the Jacobian matrix $$\begin{pmatrix}\frac{\partial (\psi \circ \phi^{-1})_1}{\partial x_1}&\dots&\frac{\partial (\psi \circ \phi^{-1}_n)}{\partial x_1}\\\vdots &\ddots & \vdots\\\frac{\partial(\psi \circ \phi^{-1})_1}{\partial x_n}&\dots& \frac{\partial (\psi \circ \phi^{-1})_n}{\partial x_n}\end{pmatrix}$$ where the $$x_j$$ are the standard coordinates in $$\mathbb R^n$$ and the $$(\psi \circ \phi^{-1})_i$$ are the coordinate functions of $$\psi \circ \phi^{-1}$$.
This is the same as in Tu if we correctly interpret his notation. He writes $$\phi = (x^1,\ldots,x^n)$$ with $$x^i : U \to \mathbb R$$ and $$\psi = (y^1,\ldots,y^n)$$ with $$y^i : V \to \mathbb R$$. We may regard the $$x^j(p)$$ and $$y^j(p)$$ as the local coordinates of the point $$p$$ with respect to the given charts.
This implies $$\psi \circ \phi^{-1} = (y^1 \circ \phi^{-1}, \ldots, y^n \circ \phi^{-1})$$, having Jacobian matrix in standard coordinates $$\begin{pmatrix}\frac{\partial (y^1 \circ \phi^{-1})}{\partial x_1}&\dots&\frac{\partial (y^1 \circ \phi^{-1})}{\partial x_1}\\\vdots &\ddots & \vdots\\\frac{\partial(y^n \circ \phi^{-1})}{\partial x_n}&\dots& \frac{\partial (y^n \circ \phi^{-1})}{\partial x_n}\end{pmatrix}$$ For local coordinates Tu defines $$\frac{\partial y^i}{\partial x^j} = \frac{\partial(y^i \circ \phi^{-1})}{\partial x_j}$$ or more precisely $$\frac{\partial y^i}{\partial x^j}(p) = \frac{\partial(y^i \circ \phi^{-1})}{\partial x_j}(\phi(p))$$ for $$p \in U \cap V$$.