Definition of dy and dx?
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Definition of dy and dx?
So I'm assuming this is a fairly basic Calc concept, but I'm still not sure I entirely get it.
dy/dx=y', the first derivative of a function y.
But dy/dx is not just a set symbol, because you can split it up by multiplying by dx.
So what exactly do the values dy and dx by themselves represent mathematically?
And if this question has been answered somewhere in these forums, please give me a link.
I tried a search, but typing dy dx in the search bar naturally pulls up a whole lot of forum links.
Thanks.
dy/dx=y', the first derivative of a function y.
But dy/dx is not just a set symbol, because you can split it up by multiplying by dx.
So what exactly do the values dy and dx by themselves represent mathematically?
And if this question has been answered somewhere in these forums, please give me a link.
I tried a search, but typing dy dx in the search bar naturally pulls up a whole lot of forum links.
Thanks.
 BlackSails
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 Joined: Thu Dec 20, 2007 5:48 am UTC
Re: Definition of dy and dx?
dx is a very small change in x. Very very small.
Re: Definition of dy and dx?
For the purposes of starting calculus, it IS just a symbol. It's convenient notation because a lot of rules work the way you might think:
eg the chain rule:
[math]\frac{dg}{dx} = \frac{dg}{df}\frac{df}{dx}[/math]
or the fundamental theorem:
[math]\int \frac{dy}{dx} dx = \int dy = y[/math]
Intuitively, they are the differences in the y and x coordinates between two very close points on the graph of the function.
After some more math, you can assign more precise meanings to these things, but that comes later.
eg the chain rule:
[math]\frac{dg}{dx} = \frac{dg}{df}\frac{df}{dx}[/math]
or the fundamental theorem:
[math]\int \frac{dy}{dx} dx = \int dy = y[/math]
Intuitively, they are the differences in the y and x coordinates between two very close points on the graph of the function.
After some more math, you can assign more precise meanings to these things, but that comes later.
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 BlackSails
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 Joined: Thu Dec 20, 2007 5:48 am UTC
Re: Definition of dy and dx?
Woods defines differentials in the following way.
The differential of an independant variable is equal to the increment of the variable; that is, dx=x_0x_1
The differential of a function y=f(x) is the principal part of the increment of y and is given by dy=f'(x)dx.
Re: Definition of dy and dx?
Thanks for the responses!
So for all calc uses for the time being, dy/dx can just be considered as a symbol? I just want to make sure I'm understand the stuff completely.
and I've seen the definition of dy=f'(x)dx in a lesson on linear approximation and differentials; unfortunately, though that may be the ultimate explanation, it just doesn't make sense to me.
So for all calc uses for the time being, dy/dx can just be considered as a symbol? I just want to make sure I'm understand the stuff completely.
and I've seen the definition of dy=f'(x)dx in a lesson on linear approximation and differentials; unfortunately, though that may be the ultimate explanation, it just doesn't make sense to me.
Re: Definition of dy and dx?
Be aware that it's somtimes written as [imath]\frac{d}{dx}y[/imath]. So that you might see something like [imath]\frac{d}{dx}(x^23x)=2x3[/imath]
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Re: Definition of dy and dx?
Don't worry, it's not just you. The notation in differential calculus is generally pretty confusing*. For now, you can just treat dy/dx, dy, dx as symbols, with the understanding that:Narius wrote:it just doesn't make sense to me.
 dx is a "small change in x"
 dy is the "corresponding small change in y"
 dy/dx is a function of x
 you can "multiply by dx or cancel the dx's" in certain cases, but this is a kind of notational shortcut (but does have a deeper meaning).
*
Spoiler:
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dubsola
dubsola
 BlackSails
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Re: Definition of dy and dx?
I like the notation. It implies the equality of mixed partials.
Re: Definition of dy and dx?
It also implies various incorrect versions of the chain rule in higher dimensions.
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 RogerMurdock
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Re: Definition of dy and dx?
Ended wrote:Narius wrote:it just doesn't make sense to me.
 you can "multiply by dx or cancel the dx's" in certain cases, but this is a kind of notational shortcut (but does have a deeper meaning).
I've been told this by multiple teachers before, but I'm not yet advanced enough to know what that deeper meaning is. I'm assuming you're talking about when you find error, you "multiply" by dx to move it to the correct side of the equation, or at least that's what it appears you're doing. Do you have any quick explanation or wikipedia links? I don't even know what to search for really.
 mmmcannibalism
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Re: Definition of dy and dx?
I look at dy and dx as delta y and delta x where the delta is very very tiny.
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Re: Definition of dy and dx?
RogerMurdock wrote:I've been told this by multiple teachers before, but I'm not yet advanced enough to know what that deeper meaning is. I'm assuming you're talking about when you find error, you "multiply" by dx to move it to the correct side of the equation, or at least that's what it appears you're doing. Do you have any quick explanation or wikipedia links? I don't even know what to search for really.Ended wrote:
 you can "multiply by dx or cancel the dx's" in certain cases, but this is a kind of notational shortcut (but does have a deeper meaning).
The dx things are called differentials. I've tried to give a sortofcoherent explanation below (spoilered for wall of text), apologies if it's pitched too low or high.
Spoiler:
Generally I try to make myself do things I instinctively avoid, in case they are awesome.
dubsola
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 RogerMurdock
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Re: Definition of dy and dx?
Ended wrote:RogerMurdock wrote:I've been told this by multiple teachers before, but I'm not yet advanced enough to know what that deeper meaning is. I'm assuming you're talking about when you find error, you "multiply" by dx to move it to the correct side of the equation, or at least that's what it appears you're doing. Do you have any quick explanation or wikipedia links? I don't even know what to search for really.Ended wrote:
 you can "multiply by dx or cancel the dx's" in certain cases, but this is a kind of notational shortcut (but does have a deeper meaning).
The dx things are called differentials. I've tried to give a sortofcoherent explanation below (spoilered for wall of text), apologies if it's pitched too low or high.Spoiler:
Thanks for the information, I understand the first approach but I struggled with the df and whatnot (I failed to see exactly what the second part had proven), but I'm going read through the wikipedia article and do a bit more research now that I know what they are called alone.
Re: Definition of dy and dx?
In the context of a calculus class I think differentials are a terrible idea. Everything you can do with them you can do with linear approximation, and much less confusingly. In the class I'm teaching right now, we did exactly that.
Re: Definition of dy and dx?
Like many things in mathematics, I did not understand the concepts of dx and dy until they were applied to Physics.
It all made sense, then  dy and dx were infinitesimals. In this context, that infinitesimals have this definition: the smallest nonzero positive number. Or, the closest you can get to zero while still having a nonzero difference.
You sort have to let this sink in for a while to get this. Think about exactly how big this has to be. And then, in that context, think of how you would compare and do math with infinitesimals.
To me, they were just a cute idea in calculus. Neat, but honestly, you never really need to understand them. So here, I shall introduce the concept of the continuous distribution, from physics, where differentials and infinitesimals dance around and play with each other and with other numbers:
(spoilered for wall of text)
Anyways, hopefully from seeing dx's and other differentials and infinitessimals being actually used and interacting with each other instead of just other normal numbers, you can start to see what they exactly are, really.
It all made sense, then  dy and dx were infinitesimals. In this context, that infinitesimals have this definition: the smallest nonzero positive number. Or, the closest you can get to zero while still having a nonzero difference.
You sort have to let this sink in for a while to get this. Think about exactly how big this has to be. And then, in that context, think of how you would compare and do math with infinitesimals.
To me, they were just a cute idea in calculus. Neat, but honestly, you never really need to understand them. So here, I shall introduce the concept of the continuous distribution, from physics, where differentials and infinitesimals dance around and play with each other and with other numbers:
(spoilered for wall of text)
Spoiler:
Anyways, hopefully from seeing dx's and other differentials and infinitessimals being actually used and interacting with each other instead of just other normal numbers, you can start to see what they exactly are, really.

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Re: Definition of dy and dx?
If you're really curious, dy and dx are differential forms. (Specifically, one forms). Don't worry if this goes way over your head. It's way over mine too, and I've been studying it.
The basic idea is that calculus is a way to talk about infinitely small changes. One of the best ways to describe an infinitely small change is to say where it starts and what direction it goes. We call this pair a tangent vector. Derivatives depend on these two pieces of information (the point you start at and the direction in which you move). In particular, derivatives should be linear functions, so we restrict ourselves to linear functions of tangent vectors. These linear functions are called cotangent vectors. Differential forms like dx, dy and df are special collections of these cotangent vectors.
Edit: To sum up, because I realized that that may not have had a very clear ending, differentials are functions that take two pieces of information as input: where you start (a point in M) and what direction you go in (a vector in M) and give you a number. This number is basically how much you change when you move a tiny (infinitely tiny, in fact) distance in that direction. The notions of tangent spaces and cotangent spaces are there so that we can talk precisely about "moving an infinitely tiny distance".
The basic idea is that calculus is a way to talk about infinitely small changes. One of the best ways to describe an infinitely small change is to say where it starts and what direction it goes. We call this pair a tangent vector. Derivatives depend on these two pieces of information (the point you start at and the direction in which you move). In particular, derivatives should be linear functions, so we restrict ourselves to linear functions of tangent vectors. These linear functions are called cotangent vectors. Differential forms like dx, dy and df are special collections of these cotangent vectors.
Spoiler:
Edit: To sum up, because I realized that that may not have had a very clear ending, differentials are functions that take two pieces of information as input: where you start (a point in M) and what direction you go in (a vector in M) and give you a number. This number is basically how much you change when you move a tiny (infinitely tiny, in fact) distance in that direction. The notions of tangent spaces and cotangent spaces are there so that we can talk precisely about "moving an infinitely tiny distance".
Last edited by MostlyHarmless on Sat Oct 17, 2009 12:12 pm UTC, edited 1 time in total.

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Re: Definition of dy and dx?
It doesn't really prove anything, it's just a way of defining things like dx so that they behave like we expect them to. But it's unnecessary for a lot of purposes.RogerMurdock wrote:Thanks for the information, I understand the first approach but I struggled with the df and whatnot (I failed to see exactly what the second part had proven)
Generally I try to make myself do things I instinctively avoid, in case they are awesome.
dubsola
dubsola
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