# Calculus of Variation: Chain Rule and Formulation Proof

• Gavroy
In summary, The conversation is about the calculus of variation and the chain rule for the gateaux derivative. The participants are looking for a proof for the chain rule and the conditions under which the derivative is equal to the partial derivative. There is also mention of a confusion regarding notation. The goal is to see the proof for the chain rule and the relationship between the gateaux derivative and the partial derivative.

#### Gavroy

I have a question about calculus of variation.

does anybody here know a proof for the chain rule:
$$\delta S= \frac{dS}{dx} \delta x$$

and for the formulation:

$$\delta S= p \delta x$$
=> $$\frac{dS}{dx}= p$$

it would be totally sufficient, if anyone here knows(e.g. a weblink) where one could see this proof.

I find your notation confusing. Are you asking the conditions under which dy/dx = [partial]y/[partial]x ?

No, it is about calculus of variation and the delta should be the gateaux derivative

Ok, if the delta S in the LHS is the $\mbox{G}\hat{\mbox{a}}\mbox{teaux}$ derivative of a functional S, then what are the things in the RHS ?

sorry that my notation confused you.

if I say $$S(x)=p \cdot x$$
then the variation says:
$$\delta S(x)=p \frac{d(x+\epsilon h)}{d \epsilon}|_{\varepsilon=0}$$
or in the other notation:
$$\delta S(x)=p \cdot \delta x$$

and now I wanted to see the proof of the chain rule:

$$\delta S(x)=\frac{\partial S}{\partial x} \cdot \delta x$$

and that:$$\frac{\partial S}{\partial x}=p$$

but I guess both proofs go hand in hand

## 1. What is the Chain Rule in the Calculus of Variation?

The Chain Rule in the Calculus of Variation is a method for calculating the derivative of a functional composed of multiple functions. It allows us to find the derivative of a functional with respect to one of its variables while holding all other variables constant.

## 2. How is the Chain Rule applied in the Calculus of Variation?

The Chain Rule is applied by taking the derivative of the outermost function and multiplying it by the derivative of the innermost function, which is then integrated with respect to the variable of interest. This process is repeated for each layer of nested functions until the desired derivative is obtained.

## 3. Can the Chain Rule be extended to functions of multiple variables in the Calculus of Variation?

Yes, the Chain Rule can be extended to functions of multiple variables in the Calculus of Variation. In this case, the partial derivatives of each variable must be taken and multiplied together in order to apply the Chain Rule.

## 4. What is the formulation proof in the Calculus of Variation?

The formulation proof is a mathematical proof that demonstrates the validity of the Euler-Lagrange equation, which is a necessary condition for a functional to have an extremum. It involves setting up a functional with a variable parameter and showing that the derivative of the functional with respect to this parameter is equal to zero at the extremum.

## 5. Why is the formulation proof important in the Calculus of Variation?

The formulation proof is important because it provides a rigorous mathematical justification for the use of the Euler-Lagrange equation in finding extrema in the Calculus of Variation. It also helps to establish the fundamental principles and concepts of the subject, making it easier to understand and apply in more complex problems.