Almost arbitrary symbolic step (calc. of var.)

[jeremusic2]

1. This is really more of a math question than a physics one... it's in calculus of variations while deriving a proof for the Euler equation. I always hit a mental road block when it goes from this step:

2. F(x,Y,Y') = I(e) --> \frac{dI}{de} = \frac{\partial F}{\partial Y} \frac{dY}{de} + \frac{\partial F}{\partial Y'} \frac{dY'}{de} *note I'm not including the integral because it's not what I'm addressing, but if you feel more comfortable with it there just use your imagination then.

3. The issue I have is why is there a sum? How do we know these variables (or really the Y and Y') are linearly independent?... or at least that's what I'm assuming it represents. I can imagine that it's possible for it to not work if they are not... idk am I missing something obvious? Or should functions and their derivatives be considered linearly independent in some function space or something?

 

[BvU]

Hello Jere and welcome to PF. Nice way to use the template, but it was intended differently.
My imagination falls short, and I don't see a sum.
Are you aware of the meaning of the partial derivative ?

 

[Chopin]

This is just the chain rule for a multi-variable function. You have to add up the contribution from each variable separately. You should be able to find a more detailed discussion of this procedure in any good multivariate calculus textbook.

 

[jeremusic2]

This is just the chain rule for a multi-variable function. You have to add up the contribution from each variable separately. You should be able to find a more detailed discussion of this procedure in any good multivariate calculus textbook.

I understand the chain rule for the most part, and I understand why normally there would be a sum - it makes sense when the two variables are independent - but now I have a Y and a Y', and I'm not convinced of why they are treated independently so that the same rules apply. I thought I made this clear in my question, I guess I didn't. I wouldn't be asking something that could be solved in two seconds by opening my calculus book. Perhaps it means I didn't acquire some crucial but subtle notion of why they can be treated independently... is it just because someone "decided" to treat them as independent variables? If so, this is allowable?

I keep looking for a direct answer to the particular question I have but it's not easy to find one. I figure it's much more efficient and effective if I ask a relative expert directly here.

 

[Chopin]

As far as F is concerned, Y and Y' are independent variables. It doesn't know that one is defined in terms of the other, or that they vary together. All it knows is that when Y varies by this much, and Y' varies by this much, it will vary by this much in response to that. So if you're looking for the response of F to a change in e, you just run the numbers like you would for any other multi-variable function.

Does that make sense? If not, maybe you could give a little more context about this equation, so that we can see in more detail how the various functions are defined.

 

[jeremusic2]

As far as F is concerned, Y and Y' are independent variables. It doesn't know that one is defined in terms of the other, or that they vary together. All it knows is that when Y varies by this much, and Y' varies by this much, it will vary by this much in response to that. So if you're looking for the response of F to a change in e, you just run the numbers like you would for any other multi-variable function.

Does that make sense? If not, maybe you could give a little more context about this equation, so that we can see in more detail how the various functions are defined.

Thanks that helped a lot, it's just weird how one can apply as you just did "as far as F is concerned". I have trouble understanding why this is ok to do just because it seems like by treating them as independent one is making a false statement. It makes sense that F can't really tell them apart though, who's to say it's not the case that anything that is treated independently actually can be dependent - just because it's treated that way doesn't mean it is I guess...

But this is why I brought up the question of the function space - are functions and their derivatives actually independent in a certain context, and is it necessary for this to be so to justify "pretending"?
For instance, the Wronskian uses the derivates of functions to test for linear independence.

 

[Chopin]

At its simplest level, a function of two variables like F just takes in two numbers and spits out a result. The fact that we label those inputs Y and Y' doesn't imply anything at all about their relationship--they're just two input parameters that you specify in order to get a value for F. It might make more sense to mentally rename them to something else like A and B, to reinforce the idea that they are just numbers. At this level, all you can do is take the partial derivative of F with respect to A or with respect to B.

Things get more interesting when the sources for A and B are functions, because like you said, they may be related to each other in some way. If both of those inputs are themselves dependent on a third variable e, then you get into the situation you listed above, where you can say $I(e) = F(Y(e), Y'(e))$. At that point, you can take the derivative of I with respect to e, for which you use the chain rule to reduce to the partial derivatives of F that we computed above: $\frac {dI}{de} = \frac{\partial F}{\partial A}\frac{dY}{de} + \frac{\partial F}{\partial B}\frac{dY'}{de}$. A change in e will in general result in a change to both Y and Y', and if that's the case, then both of them will contribute in turn to a change in F.

 

[jeremusic2]

Hello Jere and welcome to PF. Nice way to use the template, but it was intended differently.
My imagination falls short, and I don't see a sum.
Are you aware of the meaning of the partial derivative ?

When I say sum I'm just referring to the "+" sign after the arrow. And yeah I understand the meaning of the partial derivative for the most part... I think. My lack of confidence doesn't come from this at least. Here is a more detailed depiction of what step I'm talking about:

\int_{x_1}^{x_2} F(x,Y,Y')\,dx = I(ε) --> \frac{dI}{dε} = \int_{x_1}^{x_2} (\frac{\partial F}{\partial Y} \frac{dY}{dε} + \frac{\partial F}{\partial Y'} \frac{dY'}{dε})\,dx