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I am confused what differentials are and how they become related to derivatives and integrals, as neither my textbook explains them nor my teacher. What we do to solve differential equations is to convert derivative to a "differential form" like this:

$$\frac{dy}{dx}= 2x$$

$$dy = 2xdx$$

When we substitute to solve integrals, differentials also come up there.

I'm asking: How did we get differentials, what is their physical meaning (like derivative means slope, Integration results in area under curve).

I would like someone gave me a little historical aspect. When did differentials come into the scene of calculus?

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  • $\begingroup$ math.stackexchange.com/questions/925465/… $\endgroup$ – user117644 Apr 15 '15 at 10:07
  • $\begingroup$ @mistermarko My question is about differentials not how derivatives and iintegrals are related graphically. Or I'm missing something... $\endgroup$ – Haider Rehman Butt Apr 15 '15 at 10:24
  • $\begingroup$ Maybe some of the background. Anyway, the dy/dx notation is so misleading we may as well just call it 'wrong'. In IT your question would be an example of the xy problem. $\endgroup$ – user117644 Apr 15 '15 at 10:34
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Disillusioned and skeptical like you, I hated that notation and the lack of meaning to it when I first learned about it in high school. For that reason, I decided to write 80 pages about it in my Bachelor's Thesis in the History of Mathematics. The historical subject of the Leibnizian Calculus is very deep and detailed, but here I will attempt to scratch the surface:


In some textbooks $\frac{dy}{dx}$ is used as notation for $f'(x)$. In that sense the middle step below $$ \frac{dy}{dx}=\frac{g(x)}{h(y)}\iff h(y)dy=g(x)dx\iff\int h(y)dy=\int g(x)dx+c $$ really does not make much sense in itself. If $dx$ and $dy$ were never introduced formally as true entities by themself this has to be a rule for memorization purposes only.


Historically, however, $dx$ and $dy$ were given meaning individually. For any variable $t$, Leibniz defined $dt$ to denote the infinitesimal increment in $t$ when it goes from one state to the next. In that way a variable $t$ underwent consecutive stages $$ t,t+dt,(t+dt)+(dt+ddt)=t+2dt+ddt,etc. $$ with $dt+ddt$ being the next state of the (infinitesimal) variable $dt$ and $ddt$ being infinitely smaller than $dt$. One could go on forever in that way. Leibniz then put forward the principle, that on each level of being finite, infinite or infinitesimally small, the so-called order of infinity, variables that differed only by a lesser order of infinity could be considered equal.

All this had close connection to geometrical interpreations - for instance lines pointing in directions differing by an infinitesimal angle were considered to be parallel. Also, if $ds$ denoted an infinitesimal segment of a curve in variables $x,y$, then $dx^2+dy^2=ds^2$, an infinitesimal version of the Pythagorean Theorem, so that $ds$ was considered a straight line segment rather than a curve segment in that setting.

As an example, given $y=x^2$ we have $(y+dy)=(x+dx)^2$ since the next stages for $x,y$ should also satisfy the equation. Subtracting consecutive stages we then have: $$ dy=(y+dy)-y=(x+dx)^2-x^2=2x\ dx+dx^2 $$ and by Leibniz's principle $dx^2$ is infinitely smaller than the rest and can thus be disregarded.


If we should have done the same in a modern setup, we could write $y=x^2$ to have $$ \frac{\Delta y}{\Delta x}=\frac{2x\Delta x+\Delta x^2}{\Delta x}=2x+\Delta x\rightarrow 2x $$ and in that sense Leibniz's $2x\ dx+dx^2=2x\ dx$ corresponds exactly to the last step here were we go to the limit $\Delta x\rightarrow 0$. Note that $\Delta$ is used to denote actual finite differences greater than zero, whereas $d$ is used to denote infinitesimals.

I like to say that Leibniz worked permanently in the limit in some sense.


CAUTION

One should a bit catious here, though. The connection between Leibniz's calculus and modern calculus may not be as simple as the example of $y=x^2\implies dy=2x\ dx$ would suggest. Although we have the perfect connection $f'(x)=\frac{dy}{dx}$ this way, the next step $$ f''(x)=\frac{ddy}{dx^2} $$ is not always true. It is only true if we assume $ddx=0$ which requires a slightly deeper introduction to Leibniz's calculus to know what is meant by that. Also $$ h(y)dy=g(x)dx\iff\int h(y)dy=\int g(x)dx+c $$ is not trivially found to be the limit for some relation for finite differences: $$ h(y)\Delta y=g(x)\Delta x\text{ and }\sum h(y)\Delta y=\sum g(x)\Delta x $$ I once tried proving some connection for the above using the Mean Value Theorem, which worked out well but was not trivial.


As a final remark, note also that Leibniz's idea of a continuous variable increasing through distinct well defined stages has been problematized. The ontological status of infinitesimals was widely disputed in the centuries after Leibniz. A bishop, Berkeley, who was also versed in mathematics, was one of the early critics ridiculing and calling Newtons and Leibniz's infinitesimals Ghosts of Departed Quantities.

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  • $\begingroup$ For a little more detail about $ddt$ and the like and how this differs from the second order derivative of a function, see this answer I once gave to a more advanced question about the same subject. $\endgroup$ – String Apr 15 '15 at 11:36
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which textbook are you using (who is its author)? Chances are, there is a chapter or a section named "related rates of change" in your textbook, and there you can see differentials being used.

In Stewart's Calculus, to give an example, we are asked to treat derivatives as if they were a 'fraction', made up of differentials. One reason is that this kind of notation makes a lot of problems easier to follow. I believe it was Leibniz who introduced this notation, and it gradually won over the mathematical community(it was basically grafted into other notation that Newton had introduced).

Finally, when seen by itself, a differential can be understood to be a “change” in some quantity. This will be clearer when you reach the “related rates” sort of problems. In any case, things will get a LOT clearer as you work through the book! Your question is a very good one, showing that you truly want to understand what you are doing, and not just do problems “by rote”.

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  • $\begingroup$ It's Understanding pure Mathematics by A J Saddler. We do skip up a few topics. $\endgroup$ – Haider Rehman Butt Apr 15 '15 at 10:04
  • $\begingroup$ OK then in that case check out page 322 and following...does this help? $\endgroup$ – Tom Apr 15 '15 at 10:07
  • $\begingroup$ I understand how we used the Chain rule. I dont see how differentials got into derivatives and integrals. What are differentials anyway? $\endgroup$ – Haider Rehman Butt Apr 15 '15 at 10:14
  • $\begingroup$ you could say "dy" is simply a change in y, but then when you are just learning calculus in the first semester, you are given dy/dx, and told to see it as the change in y with respect to x..."seeing it from the x components' point of view". ...and differentials get a LOT more attention when you start looking at partial differential equations, because the answer will depend on which "point of view", or differential, you are working with...in any case, do you see the dfference in dy and dy/dx? $\endgroup$ – Tom Apr 15 '15 at 10:22
  • $\begingroup$ If dy is a change in y, what is the difference between dy and Δy? $\endgroup$ – Haider Rehman Butt Apr 15 '15 at 10:27

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