In the process of deriving .[tex]\nabla^2u(x,y)[/tex]. in polar coordinates I am confuse how to work through the steps. The first step from the book is to find [tex]\frac{\partial u}{\partial x} [/tex](adsbygoogle = window.adsbygoogle || []).push({});

Standard partial differentiation is at follow:

[tex]\frac{\partial u}{\partial x} = \frac{\partial u}{\partial r} \frac{\partial r}{\partial x}\;+\; \frac{\partial u}{\partial \theta}\frac{\partial \theta}{\partial x} [/tex]

If we use the same method of partial differentiation to continue to find .[tex]\frac{\partial^2 u}{\partial x^2}[/tex]. we should perform:

[tex]\frac{\partial^2 u}{\partial x^2}= \frac{\partial}{\partial r}(\frac{\partial u}{\partial x}) \;\; \frac{\partial r}{\partial x} \;+\; \frac{\partial}{\partial \theta}(\frac{\partial u}{\partial x}) \;\; \frac{\partial \theta}{\partial x} [/tex]

[tex]\Rightarrow \frac{\partial^2 u}{\partial x^2}= \frac{\partial}{\partial r}[\frac{\partial u}{\partial r} \frac{\partial r}{\partial x}\;+\; \frac{\partial u}{\partial \theta}\frac{\partial \theta}{\partial x}] \frac{\partial r}{\partial x} \;+\; \frac{\partial}{\partial \theta}[\frac{\partial u}{\partial r} \frac{\partial r}{\partial x}\;+\; \frac{\partial u}{\partial \theta}\frac{\partial \theta}{\partial x}] \frac{\partial \theta}{\partial x} [/tex]

But the book did this differently:

[tex]\frac{\partial^2 u}{\partial x^2}= \frac{\partial}{\partial x}[\frac{\partial u}{\partial r} \frac{\partial r}{\partial x}\;+\; \frac{\partial u}{\partial \theta}\frac{\partial \theta}{\partial x}] [/tex]

Then apply product rule:

[tex]\frac{\partial^2 u}{\partial x^2}= [\frac{\partial r}{\partial x}\frac{\partial}{\partial x}(\frac{\partial u}{\partial r}) + \frac{\partial u}{\partial r}\frac{\partial^2r}{\partial x^2}]\;+\; [\frac{\partial \theta}{\partial x}\frac{\partial}{\partial x}(\frac{\partial u}{\partial \theta}) + \frac{\partial u}{\partial \theta}\frac{\partial^2 \theta}{\partial x^2}][/tex]

Obvious the two don't give the same answer. Why don't the book need to perform the second derivative like the first one like what I did?

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# Confused on partial differentiation.

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