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I was reading this question Manipulating Partial Derivatives of Inverse Function when a doubt raise up. Someone can explain how the "user147263" developed the partial derivative holding $\eta-\xi$ constant? That is:

$$\xi = x - y \qquad \eta = x+y$$ $$x = \frac12(\xi+\eta) \qquad y = \frac12(\eta-\xi)$$ $$\dfrac {\partial x} {\partial \xi}\bigg|_{\eta-\xi \text{ constant}} = 1$$

How to develop the third partial derivative?

Lac
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1 Answers1

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As you want to hold $y$ constant and $y=\frac{1}{2}(\eta-\xi)$, then $(\eta-\xi)$ must be held constant.

Since $$x = \frac12(\xi+\eta)=\frac{1}{2}(\eta-\xi+2\xi)=\xi+\frac{1}{2}(\eta-\xi)$$

and $(\eta-\xi)$ is constant then it follows that $$\dfrac {\partial x} {\partial \xi}\bigg|_{\eta-\xi \text{ constant}} = 1$$

Alessio K
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