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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?

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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 $p \in U \cap V$.

See chapter "6.6 Partial Derivatives".

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