In summary, the article discusses how to calculate partial derivatives when a change of variables arises. The chain rule is applied and the inverse case is discussed.
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Introduction
A frequent concern among students is how to carry out higher order partial derivatives where a change of variables and the chain rule are involved.  There is often uncertainty about exactly what the “rules” are.  This tutorial aims to clarify how the higher-order partial derivatives are formed in this case.
Note that in general second-order partial derivatives are more complicated than you might expect.  It’s important, therefore, to keep calm and pay attention to the details.
The General Case
Imagine we have a function ##f(u, v)## and we want to compute the partial derivatives with respect to ##x## and ##y## in terms of those with respect to ##u## and ##v##.  Here we assume that ##u, v## may be expressed as functions of ##x, y##.  The first derivative usually cause no problems.  We simply apply the chain rule:
$$\frac{\partial f}{\partial x} = \frac{\partial f}{\partial u}\frac{\partial u}{\partial x} + \frac{\partial f}{\partial v}\frac{\partial v}{\partial x} \ \ \...

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This is a really helpful reference, there's so much that can go wrong with these sorts of questions and the topic is skated over in so many resources.

How common is it to have to use the inverse case? That is, if we first have a function ##F(x,y) = f(u(x,y), v(x,y))##, is it sometimes useful to restate it in the form ##G(u,v) = g(x(u,v), y(u,v))##?
 
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etotheipi said:
How common is it to have to use the inverse case? That is, if we first have a function ##F(x,y) = f(u(x,y), v(x,y))##, is it sometimes useful to restate it in the form ##G(u,v) = g(x(u,v), y(u,v))##?
It depends how the change of variables arises. For polar coordinates we generally start from:
$$x = r\cos \phi, \ \ y = r\sin \phi$$
Which is the "inverse" case in the tutorial.

The important point is that, if you have a function of ##r, \phi##, then you must calculate the inverse functions:
$$r = \sqrt{x^2 + y^2}, \ \ \ \phi = \arctan \frac y x$$
 
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It was such a nice and clear article that I really learned from it, enjoyed and admired it. The main issue is that ##f## is not directly a function of ##x## and ##y##, and I think these cases arise so much in Electrodynamics, where fields are functions of charge/current density and position, and position in itself is a function of time, even charge/current density changes with time.
 
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1. What is a second-order partial derivative?

A second-order partial derivative is a mathematical concept used to measure the rate of change of a function with respect to two different independent variables. It is calculated by taking the derivative of the first derivative of the function.

2. How do you solve second-order partial derivatives?

To solve second-order partial derivatives, you first need to find the first-order partial derivatives with respect to each independent variable. Then, you can use the chain rule to find the second-order partial derivatives by taking the derivative of the first-order partial derivatives.

3. What is the difference between a first-order and second-order partial derivative?

A first-order partial derivative measures the rate of change of a function with respect to one independent variable, while a second-order partial derivative measures the rate of change with respect to two independent variables. Essentially, a second-order partial derivative is the derivative of a derivative.

4. When do you use second-order partial derivatives?

Second-order partial derivatives are commonly used in multivariable calculus and in fields such as physics and engineering to analyze the behavior of functions with multiple independent variables. They are also used in optimization problems to find maximum or minimum values of a function.

5. Can you provide an example of solving a second-order partial derivative?

For example, if we have a function f(x,y) = x^3y^2, the first-order partial derivatives would be fx = 3x^2y^2 and fy = 2x^3y. To find the second-order partial derivatives, we would take the derivative of each of these with respect to x and y, respectively. So, fxx = 6xy^2 and fyy = 2x^3, and fxy = 6x^2y. These second-order partial derivatives can then be used to analyze the behavior of the function at specific points or to solve optimization problems.

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