MHB Finding Partial Derivatives with Transformations

[evinda]

Hello! :)

Having the transformations:
$$\xi=\xi(x,y), \eta=\eta(x,y)$$

I want to find the following partial derivatives:
$$\frac{\partial}{\partial{x}}= \frac{\partial}{ \partial{\xi}} \frac{\partial{\xi}}{\partial{x}}+\frac{\partial}{\partial{\eta}} \frac{\partial{\eta}}{\partial{x}}=\partial_{\xi} \xi_x+\partial_{\eta} \eta_x$$

$$\frac{\partial}{\partial{y}}= \frac{\partial}{ \partial{\xi}} \frac{\partial{\xi}}{\partial{y}}+\frac{\partial}{\partial{\eta}} \frac{\partial{\eta}}{\partial{y}}=\partial_{\xi} \xi_y+\partial_{\eta} \eta_y$$

$$\frac{\partial^2}{\partial{x^2}}=\frac{\partial{(\partial_{\xi} \xi_x+\partial_{\eta} \eta_x})}{\partial{x}}=\frac{\partial{(\partial_{\xi} \xi_x+\partial_{\eta} \eta_x})}{\partial{x}} \frac{\partial{\xi}}{\partial{x}}+\frac{\partial{(\partial_{\xi} \xi_x+\partial_{\eta} \eta_x})}{\partial{\eta}} \frac{\partial{\eta}}{\partial{x}}$$
I got stuck...How can I continue??

 

[I like Serena]

Hello! :)

Having the transformations:
$$\xi=\xi(x,y), \eta=\eta(x,y)$$

I want to find the following partial derivatives:
$$\frac{\partial}{\partial{x}}= \frac{\partial}{ \partial{\xi}} \frac{\partial{\xi}}{\partial{x}}+\frac{\partial}{\partial{\eta}} \frac{\partial{\eta}}{\partial{x}}=\partial_{\xi} \xi_x+\partial_{\eta} \eta_x$$

$$\frac{\partial}{\partial{y}}= \frac{\partial}{ \partial{\xi}} \frac{\partial{\xi}}{\partial{y}}+\frac{\partial}{\partial{\eta}} \frac{\partial{\eta}}{\partial{y}}=\partial_{\xi} \xi_y+\partial_{\eta} \eta_y$$

$$\frac{\partial^2}{\partial{x^2}}=\frac{\partial{(\partial_{\xi} \xi_x+\partial_{\eta} \eta_x})}{\partial{x}}=\frac{\partial{(\partial_{\xi} \xi_x+\partial_{\eta} \eta_x})}{\partial{x}} \frac{\partial{\xi}}{\partial{x}}+\frac{\partial{(\partial_{\xi} \xi_x+\partial_{\eta} \eta_x})}{\partial{\eta}} \frac{\partial{\eta}}{\partial{x}}$$
I got stuck...How can I continue??

Hey! (Mmm)

How about this (correcting a small mistake):
\begin{aligned}
\frac{\partial^2}{\partial{x^2}}

&=\frac{\partial{(\partial_{\xi} \xi_x+\partial_{\eta} \eta_x})}{\partial{x}} \\

&=\frac{\partial{(\partial_{\xi} \xi_x+\partial_{\eta} \eta_x})}{\partial{\xi}}
\frac{\partial{\xi}}{\partial{x}}
+ \frac{\partial{(\partial_{\xi} \xi_x+\partial_{\eta} \eta_x})}{\partial{\eta}}
\frac{\partial{\eta}}{\partial{x}} \\

&=(\partial_{\xi\xi} \xi_x + \partial_{\xi\eta} \eta_x) \xi_x + \partial_\xi \xi_{xx}
+ (\partial_{\eta\xi} \xi_x+\partial_{\eta\eta} \eta_x) \eta_x + \partial_\eta \eta_{xx}\\

\end{aligned}

 

[evinda]

Hey! (Mmm)

How about this (correcting a small mistake):
\begin{aligned}
\frac{\partial^2}{\partial{x^2}}

&=\frac{\partial{(\partial_{\xi} \xi_x+\partial_{\eta} \eta_x})}{\partial{x}} \\

&=\frac{\partial{(\partial_{\xi} \xi_x+\partial_{\eta} \eta_x})}{\partial{\xi}}
\frac{\partial{\xi}}{\partial{x}}
+ \frac{\partial{(\partial_{\xi} \xi_x+\partial_{\eta} \eta_x})}{\partial{\eta}}
\frac{\partial{\eta}}{\partial{x}} \\

&=(\partial_{\xi\xi} \xi_x+\partial_{\xi\eta} \eta_x) \xi_x
+ (\partial_{\eta\xi} \xi_x+\partial_{\eta\eta} \eta_x) \eta_x \\

\end{aligned}

According to my textbook it is like that:
$$\frac{\partial^2}{\partial{x^2}}=\frac{\partial{(\xi_x \partial_{\xi}+\eta_x \partial_{\eta})}}{\partial{x}}=\xi_{xx} \partial_{\xi}+\xi_x \partial_x \partial_{\xi}+\eta_{xx} \partial_{\eta}+\eta_x \partial_x \partial_{\eta}$$
Why is it so?

 

[I like Serena]

According to my textbook it is like that:
$$\frac{\partial^2}{\partial{x^2}}=\frac{\partial{(\xi_x \partial_{\xi}+\eta_x \partial_{\eta})}}{\partial{x}}=\xi_{xx} \partial_{\xi}+\xi_x \partial_x \partial_{\xi}+\eta_{xx} \partial_{\eta}+\eta_x \partial_x \partial_{\eta}$$
Why is it so?

Let's apply the sum rule and the product rule:
\begin{aligned}
\frac{\partial{(\xi_x \partial_{\xi}+\eta_x \partial_{\eta})}}{\partial{x}}

&= \frac{\partial(\xi_x \partial_{\xi})}{\partial{x}} + \frac{\partial(\eta_x \partial_{\eta})}{\partial{x}} \\

&= \xi_{xx} \partial_{\xi} + \xi_x \partial_x\partial_\xi
+ \eta_{xx} \partial_{\eta} + \eta_x \partial_x\partial_\eta\\

\end{aligned}
I worked it out completely, but apparently that is not what was asked!
Ah well, since I have already written it, I'll leave it here.

So let's suppose we have a function $f(\xi, \eta)$.

Then:
$$\frac{\partial f}{\partial x}
= \frac{\partial f}{\partial \xi} \frac{\partial \xi}{\partial x}
+ \frac{\partial f}{\partial \eta} \frac{\partial \eta}{\partial x}$$

Applying both the chain rule and the product rule:
\begin{aligned}
\frac{\partial^2 f}{\partial x^2}

&= \frac{\partial}{\partial x}
\left( \frac{\partial f}{\partial \xi} \frac{\partial \xi}{\partial x}
+ \frac{\partial f}{\partial \eta} \frac{\partial \eta}{\partial x} \right) \\

&= \frac{\partial}{\partial x} \left( \frac{\partial f}{\partial \xi} \right)
\frac{\partial \xi}{\partial x}
+ \frac{\partial f}{\partial \xi}
\frac{\partial}{\partial x}\left( \frac{\partial \xi}{\partial x} \right )

+ \frac{\partial}{\partial x} \left( \frac{\partial f}{\partial \eta} \right)
\frac{\partial \eta}{\partial x}
+ \frac{\partial f}{\partial \eta}
\frac{\partial}{\partial x}\left( \frac{\partial \eta}{\partial x} \right) \\

&= \left( \frac{\partial^2 f}{\partial \xi^2}\frac{\partial \xi}{\partial x}
+ \frac{\partial^2 f}{\partial\eta\partial\xi}\frac{\partial \eta}{\partial x} \right)
\frac{\partial \xi}{\partial x}

+ \frac{\partial f}{\partial \xi}
\frac{\partial^2 \xi}{\partial x^2}

+ \left( \frac{\partial^2 f}{\partial \xi\partial\eta} \frac{\partial \xi}{\partial x}
+ \frac{\partial^2 f}{\partial \eta^2} \frac{\partial \eta}{\partial x} \right)
\frac{\partial \eta}{\partial x}

+ \frac{\partial f}{\partial \eta}
\frac{\partial^2 \eta}{\partial x^2} \\

&= f_{\xi\xi}\xi_x^2 + f_{\eta\xi}\eta_x\xi_x
+ f_\xi \xi_{xx}
+ f_{\xi\eta} \xi_x\eta_x + f_{\eta\eta} \eta_x^2
+ f_\eta \eta_{xx} \\

&= f_{\xi\xi}\xi_x^2 + 2 f_{\eta\xi}\eta_x\xi_x + f_{\eta\eta} \eta_x^2
+ f_\xi \xi_{xx} + f_\eta \eta_{xx} \\

&= \left(\ \xi_x^2\partial_\xi\partial_\xi + 2 \eta_x\xi_x\partial_\eta\partial_\xi
+ \eta_x^2 \partial_\eta\partial_\eta
+ \xi_{xx}\partial_\xi + \eta_{xx}\partial_\eta \ \right)\ f \\

\end{aligned}