Finding Partial Derivatives with Transformations

In summary, we can find the partial derivatives of a function $f(\xi, \eta)$ by using the chain rule and the product rule, and we can find the second derivative with respect to $x$ by applying the same rules twice and using the given transformations $\xi=\xi(x,y)$ and $\eta=\eta(x,y)$.
  • #1
evinda
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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??
 
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  • #2
evinda said:
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}
 
  • #3
I like Serena said:
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? :confused:
 
  • #4
evinda said:
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? :confused:

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! :eek:
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}
 

FAQ: Finding Partial Derivatives with Transformations

What is a partial derivative?

A partial derivative is a mathematical concept used in multivariate calculus to describe the instantaneous rate of change of a function with respect to one of its variables while holding all other variables constant. It is denoted by ∂ (pronounced "del") followed by the variable with respect to which the derivative is taken.

What is a transformation?

A transformation is a mathematical operation that changes the coordinates or values of a function. In the context of finding partial derivatives, transformations can be used to simplify the process by changing the variables to ones that are easier to work with.

Why do we need to use transformations when finding partial derivatives?

Transformations can help us simplify the process of finding partial derivatives by changing the variables to ones that are easier to work with. This can save time and make the calculations more manageable.

How do we use transformations to find partial derivatives?

To use transformations when finding partial derivatives, we first substitute the transformed variables into the original function. Then, we calculate the partial derivatives with respect to the transformed variables. Finally, we substitute the original variables back in to get the final partial derivatives.

What are some common transformations used in finding partial derivatives?

Some common transformations used in finding partial derivatives include changing to polar coordinates, using logarithmic or exponential transformations, and using trigonometric transformations such as sine and cosine. The specific transformation used will depend on the variables and function being worked with.

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