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I am trying to differentiate a function $f(x,y,\frac{\partial y}{\partial x},\frac{\partial(\frac{\partial y}{\partial x})}{\partial x})$ w.r.t $g(x,y)$ but I am not quite sure how to proceed. I tried following the explanation here- Derivative of $f(x,y)$ with respect to another function of two variables $k(x,y)$ but I am having a difficulty expanding the idea to my case. In the above case, consider $x$ to be the independent variable.

It would be great if you could also suggest a reference book I can follow.

EDIT to show specific problem:

I have a function - F In this n is defined as n = ($cos\theta$$cos\phi$,$cos\theta$$sin\phi$,$sin\theta$). Further, $\theta$ = $\theta(x,y,z)$ and $\phi$ = $\phi(x,y,z)$. I want to evaluate Derivative.

I am presently simulating the flow of nematic liquid crystals using Leslie Ericksen theory. I am adding a screenshot of all the equations of the theory. Essentially, n describes the orientation of a liquid crystal. Depending on the external forces ( may be a flow or magnetic field or an electric field) change orientation to obtain a state with the lowest energy (most stable). Summary of all equations.

In my actual problem, $(x,y,z)$ are the cartesian coordinates and the others are functions that dependent on them.

SomBhat
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    What does that even mean? – copper.hat Feb 01 '18 at 04:21
  • I want to calculate $\frac{\partial f}{\partial g}$ where f is a function of n variables and g is a function of m variables – SomBhat Feb 01 '18 at 18:29
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    I can read what you wrote, but I have still no idea what it means. – copper.hat Feb 01 '18 at 18:31
  • I have edited my question to include the exact problem. – SomBhat Feb 01 '18 at 20:02
  • I am familiar with derivatives in the usual sense, but there is some context to your problem that I am missing. – copper.hat Feb 01 '18 at 20:05
  • Can you please tell me what sort of information you are looking for? – SomBhat Feb 01 '18 at 23:28
  • A definition of what it means. – copper.hat Feb 01 '18 at 23:33
  • Added the physical basis. I do not know what else I can add – SomBhat Feb 02 '18 at 18:16
  • @copper.hat You are inverting $g$ to obtain new coordinates $y$, then write $f$ in those coordinates and differentiate with respect to $y$. – Pedro Feb 02 '18 at 18:20
  • @copper.hat, do you have any ideas for the problem I have mentioned? – SomBhat Feb 05 '18 at 17:41
  • @SomBhat: I don't understand what you are trying to do. I don't understand the notation (what is ${\partial y \over \partial x^2}$?). I don't know what $g$ is (what is the range?) – copper.hat Feb 05 '18 at 17:46
  • @copper.hat, I am sorry, that is meant to be the second derivative of y w.r.t x. For my case g is a non-negative real quantity. – SomBhat Feb 06 '18 at 18:56
  • @SomBhat: Sorry, I don't understand what you are trying to do :-(. – copper.hat Feb 06 '18 at 19:35
  • You will need to be a bit more specific, I think. There is a sense by which you can differentiate with respect to a function that is not naturally one of the independent variables, if the independent variables are related in some way to the variable with respect to which you want to differentiate. – Ian Feb 06 '18 at 19:53
  • (Cont.) For example, if you have $E(S,V)$ and $T=\left ( \frac{\partial E}{\partial S} \right )_V$ (where the subscript means "hold these variables constant"), you can make sense of $\left ( \frac{\partial E}{\partial T} \right )_V$. But it is crucial that you fully specify what is held constant and what isn't. In your case it doesn't seem that you have done so sufficiently. – Ian Feb 06 '18 at 19:54
  • @Ian is there more info required for clarity? – SomBhat Feb 07 '18 at 03:29
  • @SomBhat Yes; basically, to differentiate with respect to $g$, you move in some path in the underlying $(x,y)$ space. But you need to say what that path is, which means holding at least one quantity (usually exactly one quantity) constant. – Ian Feb 07 '18 at 13:54
  • @Ian, I finally understood what you mean. Referring to the problem I have shown in the Edit, I am evaluating the derivative at a given coordinate $(x,y,z)$. I don't think it is possible to hold all 3 constant (or is it)? – SomBhat Feb 08 '18 at 20:23

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