Why does the multi variable chain rule need to be proven?

[Gauss M.D.]

It seems to me to follow from the definition of a partial derivative.

If f(x,y) = f_x + f_y, then what else can Δf be other than ∂f/∂x*Δx + ∂f/∂y*Δy? Then in the limiting case, the changes in f, x and y becomes differentials instead. All this seems to be given by the definitions themselves, is it not?

So we have

df = ∂f/∂x*dx + ∂f/∂y*dy

All that's left is dividing by the parameter differential dt and we have our chain rule. I seem to recall there being entire lectures devoted to 'proving' this however. What am I missing?

 

[arildno]

What you've written is intuitive hand-waving of no rigour.
Just because something strikes you as "obvious", doesn't mean that it is proven.

 

[Gauss M.D.]

I'm not hand waving, I'm referring to the definition of a partial derivative. Definitions doesn't need proving, right?

By "If f(x,y) = fx + fy, then what else can Δf be other than ∂f/∂x*Δx + ∂f/∂y*Δy?", I'm saying that it follows from the definition that a change in f to be however much we change x by times the rate of change of f with respect to x, plus the change in y times the rate of change of f with respect to y. That's just vectorial addition.

 

[Office_Shredder]

By "If f(x,y) = fx + fy, .

I don't understand this claim. Are you saying f can be written as the sum of a function depending only on x and a function depending only on y?

 

[Gauss M.D.]

No I was referring to a vector valued function with a î and a ĵ component, which I guess was a mistake. I don't think it's a necessary condition for my 'claim' though?

 

[Gauss M.D.]

Maybe I'm phrasing it incorrectly. Can we pose this question instead:

If we define the partial of f wrt x_i as the rate of change in f wrt x_i, keeping everything but x_i constant, in what other way can we define a change in f other than the sum of all ∂f/∂x_i*Δx_i? What part of this definition lacks rigour?

 

[Office_Shredder]

The part where you say "In what other way?". The answer is that I don't know what other ways we could do this, but if you want to be rigorous you have to prove that. If someone said "in what other way could we model the real numbers?" you'd probably be stumped, but it turns out there's a whole branch of mathematics devoted to studying alternative models of the real numbers

As one more concrete example, you seem to be assuming that if there is a small change in x, it does not have an appreciable effect on what a small change in y will do to my function (and vice versa), without even stating that it's a point of interest, much less proving it to be the case.

 

[UltrafastPED]

The generalized chain rule is not hard to prove, but there is a lot of bookkeeping.

Here is a proof of the simple chain rule: http://math.rice.edu/~cjd/chainrule.pdf

Here is a proof of the general chain rule for two variables: http://math.ucsd.edu/~ashenk/Section14_4.pdf

You can see that the bookkeeping increases ...

 

[Gauss M.D.]

The part where you say "In what other way?". The answer is that I don't know what other ways we could do this, but if you want to be rigorous you have to prove that. If someone said "in what other way could we model the real numbers?" you'd probably be stumped, but it turns out there's a whole branch of mathematics devoted to studying alternative models of the real numbers

As one more concrete example, you seem to be assuming that if there is a small change in x, it does not have an appreciable effect on what a small change in y will do to my function (and vice versa), without even stating that it's a point of interest, much less proving it to be the case.

Good point with the second paragraph, I guess that isn't self evident at all. If you don't mind me asking, is there such functions, exhibiting the behaviour you're talking about or are you just making a point?

 

[Mandelbroth]

It seems to me to follow from the definition of a partial derivative.

If f(x,y) = f_x + f_y, then what else can Δf be other than ∂f/∂x*Δx + ∂f/∂y*Δy? Then in the limiting case, the changes in f, x and y becomes differentials instead. All this seems to be given by the definitions themselves, is it not?

So we have

df = ∂f/∂x*dx + ∂f/∂y*dy

All that's left is dividing by the parameter differential dt and we have our chain rule. I seem to recall there being entire lectures devoted to 'proving' this however. What am I missing?

Differentials aren't "limiting cases" of changes in a variable. Differential geometry gives a more formal meaning to differentials and differential forms, but for now it should suffice for you to know that what you have said is mostly nonsensical.

The multivariate chain rule needs to be proved because it is important to the understanding of multivariate calculus.

 

[Jorriss]

Gauss MD, this might be of interest.
https://www.dpmms.cam.ac.uk/~wtg10/meanvalue.html
It's not about the chain rule but it might be insightful as to why proofs are necessary and why what seems obvious, is not so obvious.

 

[Office_Shredder]

Good point with the second paragraph, I guess that isn't self evident at all. If you don't mind me asking, is there such functions, exhibiting the behaviour you're talking about or are you just making a point?

Consider the function

f(x,y) = xsin(1/y) for y not equal to zero
f(x,0) = 0 for all x.

Then let's look at f(0,0). If I jiggle just x a little bit, my function is identically zero, so the partial derivative in the x direction is zero. If I jiggle just y a little bit, my function is identically zero, so the partial derivative in the y direction is zero. However, if I jiggle x a little bit, then the function

g(y) = xsin(1/y) for y not zero, and g(0) = 0

is not even a continuous function, so a little change in y is going to cause a change of magnitude up to size x, regardless of how small my change in y is.

 

[Gauss M.D.]

Differentials aren't "limiting cases" of changes in a variable. Differential geometry gives a more formal meaning to differentials and differential forms, but for now it should suffice for you to know that what you have said is mostly nonsensical.

The multivariate chain rule needs to be proved because it is important to the understanding of multivariate calculus.

Excellent attitude, I'll just creep back into my dumb person hole then. Apologies for the curiosity.

 

[Gauss M.D.]

(Most of you should also note that the question wasn't "why does anything that makes intuitive sense geometrically need proving ever", which I suppose is a tiresome question, but "why doesn't this follow from the definition".)

 

[Mandelbroth]

Excellent attitude, I'll just creep back into my dumb person hole then. Apologies for the curiosity.

Thank you. I take my attitude to be very fun.

The point of this is to say that what you have written is wrong, not that you are stupid. Don't be oversensitive to what we say in that regard. You're still learning (and so am I!) so it's okay to make mistakes. I make plenty of mistakes (you can see plenty of them here and here).

Given, my dumb person hole has 10 2-liter bottles of Dr. Pepper (my Muddah loves me), so I think I'll go back there.

 

[WannabeNewton]

The problem is that what you're stating is not a definition. It is, as many have pointed out, just hand-waving "physics" logic, especially the sinful act of dividing by differentials.

 

[lurflurf]

I depends on specifics. If the single variable chain rule is carefully formulated (ie not as in most calculus books) the multiple variable chain rule is obvious as the same proof can be used with vector variables in place of real numbers. Certain definitions of partial derivatives can help as well, but I think using cute definitions obscures the idea.