I don't understand leibniz notation

[DHS Science]

Leibniz notation is made of ratios of differential operators? right? what does this mean? What is a differential operator? Why can we take this ratio? In a u subsitution, why can we break apart du/dx? this doesn't make sense!

 

[complexPHILOSOPHY]

$$\frac{dy} {dx}$$ signifies that you are taking the derivative of $$f(x)$$ where $$y=f(x)$$ and the demoninator $$dx$$ indicates the variable for differentiation.

According to the Chain Rule:

$$\frac{dy} {dx}=\frac {dy} {du}\cdot \frac {du} {dx}\cdot$$

It is to indicate an operation which is why it is not considered a ratio. There are other notations such as Lagrange’s and Newton’s.

Now, an author or lecturer can treat this notation differently, I believe and perhaps ‘define’ specific variables (which may be why you can break $$\frac {du} {dx}$$ apart, but I am not certain), although I am not sure about that as I am still teaching myself Calculus as well. If I made a mistake, feel free to correct me I am still building a theoretical framework of Calculus and I have been working in abstract algebra lately and need to start working through Calculus again.

 

[ObsessiveMathsFreak]

Dispite centuries of work, the notation for derivatives is still worfully substandard. This is complicated by the fact that very often when a derivative is written down the author may mean one of two different but still isomorphic things.

We've got a function $$y(x)$$. Sometimes you'll see the derivative written in one of two ways.

Sometimes you have the derivative written as $$\frac{d}{dx}\left(y\right)$$ where in this case what we means is the operation of differenciation applied to the function y(x) to produce some other function, y'(x), which we take to be the derivative of y.(whatever that is). This notation is often written as $$\partial_x y$$ or $$D_x y$$ to get rid of the fraction altogether.

More often you will see the derivative denoted as $$\frac{dy}{dx}$$, which for all the world looks like some number dy divided by another number dx. Most nowadays regard this second version as being shorthand or interchangable with the first and meaning the exact same thing, with any similarity to a fractions arising soley from typographical considerations.

This dismissal is actually completely wrong. Well, not in the context of what the author means. They mean $$\frac{d}{dx}\left(y\right)$$. But of course, thought they mean what they say, they very often do not say what they mean. They say $$\frac{dy}{dx}$$. And this really does mean the infinitesimal dy divided by the infinitesimal dx. Whatever an infinitesimal is.

Leibniz created $$\frac{dy}{dx}$$ like a fraction because he meant that it really was a fraction, or a ratio if you like, of two "infinitesimals". Both were less than any positive number but were not equal to zero. This is honestly what he meant. He also experimented with the notation dy:dx amoung others. But as it stands $$\frac{dy}{dx}$$ really does mean a ration of rates of change, whatever those are.

Now, well, sometimes to doesn't. A lot of people regard $$\frac{dy}{dx}$$ as a holdover and simply this of the functional operation of differenciation, or else some limiting process. This will be implicit in most (but not all!) papers and books nowadays. My understanding is that recently someone has made the concept of infinitesimals rigorous, which confirms everyone's intuition all along, that they were legitimate concepts. Most don't use this point of view though.

Your difficulties with the notation of the derivative are no accident or failing on your part. The notation of differential equations is really not very satisfactory in general. For example, if you were to see $$\frac{dy}{dx}(2x)$$, does that mean y'(2x) or 2y(x)? What do such enigmas as $$\frac{\partial}{\partial q}\left(\frac{d}{dt}\left(\frac{\partial L(q,\dot{q},t)}{\partial \dot{q}}\right)\right)$$ actually mean? The algebra of calculus is not as clear as we might like.

 

[complexPHILOSOPHY]

Well ObsessiveMathsFreak definitely taught me a few things as well. :)

Abraham Robinson introduced infinitesimal calculus using hyperreals, I believe (if this text is giving the proper individual credit).

http://www.math.wisc.edu/~keisler/calc.html

[EDIT: I found the PDF in the Math Tutorials Section ]

 

[arildno]

Why the $\partial$?
A much better, and relatively standard notation is $y_{x}$

This is readily extended to partial differentiation with variables held constant shown explicitly (common in thermodynamics, for example):
If y is a quantity depending on quantities x,u, v, the derivative of y wrt. to x may be written:
$$y_{x,uv}$$

 

[HallsofIvy]

The crucial point, and the reason why dy/dx notation is so widely used is that while dy/dx is not a fraction, it can always be treated as one. The derivative is the limit of a fraction- in proving the "chain rule", dy/dx= dy/dt dt/dx, or the "inverse derivative rule", dx/dy= 1/(dy/dx), we can always go back "before" the limit, use the fraction property, then take the limit again. "Differential notation" was developed spefically to make use of that fact.

 

[Gib Z]

arildno: The $\partial$ makes your work look harder to onlookers :P

 

[DHS Science]

The crucial point, and the reason why dy/dx notation is so widely used is that while dy/dx is not a fraction, it can always be treated as one. The derivative is the limit of a fraction- in proving the "chain rule", dy/dx= dy/dt dt/dx, or the "inverse derivative rule", dx/dy= 1/(dy/dx), we can always go back "before" the limit, use the fraction property, then take the limit again. "Differential notation" was developed spefically to make use of that fact.

so are you saying that dy/dx is not the ratio, but rather the limit of the ratio? Shorthand, if you will, for the limit definition of a derivative? If this is the case, then why can you split the ratio (like in a u subsitution) AFTER you take the derivative? I know the proof for the chain rule, but I don't really understand it because it still doesn't really make sense to me that you can reduce an operation.

Leibniz created [dy/dx] like a fraction because he meant that it really was a fraction, or a ratio if you like, of two "infinitesimals". Both were less than any positive number but were not equal to zero.

Are you saying basically the same thing? That these infinitesimals are the limits of the change in y over the change in x as the change approaches 0? Is an infinetesimal just an infinetly small number?

If all dy/dx is is shorthand for the limit version of a derivative, then I have even less of an idea about what a differential operator is. I've been told that this will somehow magically make sense when I get to linear algebra, but I have a ways to get there as I'm only in calc 2 now and this is driving me crazy.

 

[DHS Science]

As I'm thinking about this... Leibniz notation can't just be shorthand for the limit definition of a derivative because the limit of the change in y over the limit of the change in x is not equal to the limit of the change in y over the change in x

 

[Hurkyl]

They say $$\frac{dy}{dx}$$. And this really does mean the infinitesimal dy divided by the infinitesimal dx. Whatever an infinitesimal is.

That may have been what people meant in the early 18th century, but we're in the 21st century now. You're a few years behind the times. :tongue:

My understanding is that recently someone has made the concept of infinitesimals rigorous, which confirms everyone's intuition all along, that they were legitimate concepts. Most don't use this point of view though.

You're probably referring to nonstandard analysis. In nonstandard analysis, dy(x)/dx is still not a ratio of infinitessimals:

$$\frac{d y(x)}{dx} := \mathop{\text{std}} \left( \frac{{}^\star y(x + \epsilon) - {}^\star y(x)}{\epsilon} \right)$$

iff the expression on the r.h.s. has exactly the same value for every nonzero infinitessimal $\epsilon$. ${}^\star y$ means the nonstandard version of the function y, and std means to round to the nearest standard number.


I know of two other notions that "infinitessimal" would be an appropriate description, one in standard analysis, and one in algebra, and in both of them, the derivative is still not given by a ratio of infinitessimals. (In fact, in neither of them can you divide by an infintiessimal)

 

[Gib Z]

Yes, but newbies who want an easy intuitive approach don't give a damn about rigor. They think:

Gradient Equals = change in y/change in x.

As we take the limit of the changes to zero, rewrite the small triangles as d's.

Derivative = dy/dx.

Tiny Length, change in y, is dy. Tiny length, change in x, dx. Tiny lengths still do exist, and are different entites. They can be seperated, dy/dx is merely a fraction containing the 2.

They do not bother to define infintessimal, they just think its a really small number.