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I need some help with basic calculus. I asked a question the other day and got a decent answer but there is one step in the answer I just don't understand. Why is ${\partial y_1 \over \partial x_1} $ positive? Here $y_i$ and $x_i$ are charts and we are on a manifold with boundary so that $x_1=y_1=0$ on the boundary.

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Let's consider an example: Take $M = \{x\in\mathbb R^2: x_1\ge 0\}$. Take a different chart $(y_1,y_2)$, say at the origin, with the boundary given by $y_1=0$. This means that $y_1(0,x_2)=0$ and $y_1(x_1,x_2)>0$ for $x_1>0$. By the limit definition of the derivative, we'll have $\frac{\partial y_1}{\partial x_1}(0,0)\ge 0$. But it can't be $0$: The Jacobian determinant must be nonzero (by definition of chart) and $\frac{\partial y_1}{\partial x_2}(0,0)=0$.

I leave it to you to do the $n$-dimensional case.

Ted Shifrin
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