# Question on Calculus of variations formalism

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• guitarphysics

#### guitarphysics

Hi all, I had a quick question regarding the formalism behind calculus of variations. In one-dimensional standard calc, we consider functions $$f:\mathbb{R}\to \mathbb{R}$$ and define their derivatives using the conventional definition with the limit of the quotient of the change in the function, and the change of the function's parameter.

When we move on to calculus of variations, we instead consider functions (which we for some reason call functionals) $$F:\mathcal{F}(\mathbb{R})\to\mathbb{R}$$ where $\mathcal{F}(\mathbb{R})$ is the set of all functions that map $\mathbb{R}\to\mathbb{R}$. We define the derivative of these functions in an analogous way to how we did for real one-dimensional real functions. (And usually we'll be interested in finding stationary points, but that's another story.) Is this basically it?

I'm posting this because 1) I was confused because in physics textbooks, I recall reading things like "calculus of variations is about minimizing functionals, which are different from functions". But I'm trying to check to see if what they actually meant is that functionals are different from real functions (since it seems like functionals are just another specific type of function). And 2) because initially calculus of variations seemed like an especially hard thing, postponed for later in physics education when one was ready for things like Lagrangian mechanics; but it's starting to look like just another version of calculus, using a different set of functions to study! So I just wanted to check that if it is this straightforward :)

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