Chain rule - legendre transformation

In summary, when taking partial derivatives, it is important to specify which variables are being held constant. In this conversation, the focus was on calculating (∂g/∂x)|q and there were two ways to approach it. The first was to use the differential of g and set dq=0, which results in (∂g/∂x)|q = p. The second was to use the method of taking partial derivatives, but it is important to note that (∂f/∂x)|q ≠ p and instead (∂f/∂x)|y = p. It is also possible to manipulate the expression to reach the same conclusion as the first method. Additionally, it is not necessary to assume that y
  • #1
sgh1324
12
0
let

df=∂f/∂x dx+∂f/∂y dy and ∂f/∂x=p,∂f/∂y=q

So we get

df=p dx+q dy

d(f−qy)=p dx−y dqand now, define g.

g=f−q y
dg = p dx - y dq

and then I faced problem.

∂g/∂x=p←←←←←←←←←←←←←←← book said like this because we can see g is a function of x and p so that chain rule makes ∂g/∂x=p

but I wrote directly ∂g/∂x so i can get a result,

∂g/∂x=∂f/∂x−y ∂q/∂x−q ∂y∂x=p−y ∂q∂x - 0(y is independent of x)

I know that variation y is independent of x, but I'm not sure that q is also the independent function of x. what if the func f is xy? if I set like that,

g=f−qy=xy−xy=0

it's not same with p!mathematical methods in the physical sciences, Mary L. Boas 3rd edition page231
http://www.utdallas.edu/~pervin/ENGR3300/Boaz.pdf
 
Physics news on Phys.org
  • #2
In taking partial derivatives , it is recommended to explicitly show which variable is maintained constant. So you want to calculate (∂g/∂x)|q. There are two ways. The easiest one, is to look at the differential of g and set dq=0 since you are holding q constant. This way you get (∂g/∂x)|q = p. The second is to use your method. Then,

(∂g/∂x)|q=(∂f/∂x)|q - q (∂y/∂x)|q --------- (1)

notice that (∂f/∂x)|q ≠ p. Correct is (∂f/∂x)|y = p.

It is possible to manipulate expression(1) above to reach to the same conclusion that we obtain easily from the differential which is (∂g/∂x)|q = p.

Also note that there no need to assume that y is independent of x. In fact in the context of applying Legendre Transform to Thermodynamics, the natural variables do depend on each other. For example U=(S,V,N) where U is the internal energy, and we have a dependence of S on N and V.
 

Related to Chain rule - legendre transformation

What is the Chain Rule?

The Chain Rule is a mathematical principle that allows us to find the derivative of a composite function. It states that the derivative of a composite function is equal to the product of the derivative of the inner function and the derivative of the outer function.

What is a Legendre Transformation?

A Legendre Transformation is a mathematical tool used to transform one set of variables into another, typically in order to simplify a problem or express it in a different form. In particular, it is used to transform between the Lagrangian and Hamiltonian formulations in classical mechanics.

What is the relationship between the Chain Rule and Legendre Transformation?

The Chain Rule and Legendre Transformation are closely related because both concepts involve transforming one set of variables into another. In the case of the Chain Rule, this transformation is used to find the derivative of a composite function. In the case of the Legendre Transformation, it is used to transform between different formulations of a problem.

How is the Chain Rule used in the Legendre Transformation?

In the Legendre Transformation, the Chain Rule is used to find the derivatives of the original variables with respect to the new variables. This allows us to express the original problem in terms of the new variables, making it easier to solve.

What are some real-world applications of the Chain Rule and Legendre Transformation?

The Chain Rule and Legendre Transformation have many applications in physics and engineering. They are used to solve problems in classical mechanics, thermodynamics, and fluid dynamics, among others. They are also used in economic and financial modeling to simplify complex problems and make them easier to analyze.

Similar threads

Replies
6
Views
2K
Replies
4
Views
286
Replies
9
Views
2K
Replies
1
Views
1K
  • Calculus
Replies
5
Views
1K
Replies
7
Views
2K
Replies
5
Views
2K
Replies
5
Views
1K
Replies
1
Views
1K
Back
Top