# What is the formal definition of $d$, or $\partial$, in differation and integration

This might sound a bit like a silly question, but i'm a second year math student, and so far i've encountered $d$ or $\partial$ in many cases ofcourse (mostly in calculus :)). Those letters or symbols came in many forms, when derivating and integrating, e.g:

$\frac{df}{dx},\,\int g\left(x\right)dx,\,\frac{\partial^{2}h}{\partial y^{2}}$

I can work with them, and i'm capable of derivating and integrating with ease. However, whenever i ask a professor about a formal definition, i get an answer of the sort: "Consider this to be some sort of symbol, we won't define this yet".

Those things must have formal definitions, as some actions can be performed on them, they sometime cancel eachother, etc...

So, can anyone give me something with meat?

Thanks!

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These things are defined properly in any good introduction to real analysis, which I'm sure you will come to at some point in your mathematical studies. –  Zhen Lin May 25 '11 at 10:39
@IBSL Doesn't wikipedia help? –  user9413 May 25 '11 at 10:42
@Chandru: found definitions for derivatives, and for integrals. but not for those symbols... link? –  IBS May 25 '11 at 10:47
@IBS: Wikipedia total derivative –  user9413 May 25 '11 at 10:55
Another perspective is that of differential forms: f is considered a 0-form, you get a 1-form by applying the d- operator, if you are familiar with this perspective. Otherwise, df is the change in the value of f along the linear approximation f'(x) to the function. In 1-d, the approximation is a line; the actual change in values of f is approximated by the change along the tangent line: in your curve y=$x^2$, the line y=2x locally aproximates your function as precisely as you want in a precise sense. Then d$x^2$ is 2xdx because this is the change in the values of the approximating line y=2x. –  gary May 25 '11 at 11:20

## 2 Answers

Formally, the letter d doesn't carry one single meaning in these expressions, it's just a part of the notation. However, there's obviously a deeper reason for that notation. Informally, you can think of $\mathrm dx$ as a "small" change in $x$, representing a limit where the change $\Delta x$ goes to $0$, but the limiting processes required to make this rigorous can be different in different cases. In your first two examples, they are

$$\frac{f(x+\Delta)-f(x)}{\Delta x}=\frac{\Delta f}{\Delta x}\to \frac{\mathrm df}{\mathrm dx}$$

and

$$\sum f(x)\Delta x\to\int f(x)\mathrm dx\;,$$

where the latter limit takes some care to define properly (see Riemann integral). That you can often "cancel" expressions involving the d (for instance in the substitution rule for integrals and in the chain rule for derivatives) is due to this common origin in limiting processes for small changes, but this needs to be proved in each case.

A more rigorous version of the concept of "small changes" is the differential of a function. But formally the notations in your examples don't refer to such differentials; as mentioned, the letter d in them is just a fixed part of the notation.

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Dear IBS, $dx$, $dy$ are just differentials of $x$ and $y$ respectively, so they're really $\Delta x=x_{\rm after}-x_{\rm before}$ and $\Delta y = y_{\rm after} - y_{\rm before}$ assumed to be infinitesimally small. The ordinary derivative $dy/dx$ is literally the radio of $\Delta y / \Delta x$ in the limit when $\Delta x$ is small.

The integral $\int dx\,f(x)$ is literally the sum (the sign $\int$ is supposed to be a mutated letter "S" that stands for "sum") of the products of $f(x)$ - height of a thin rectangle - and the infinitesimal variations $dx$ - the width of the thin rectangle - that have exactly the same meaning as in the previous paragraph - $\Delta x$ expected to be infinitesimally small. The notation is particularly natural for the Riemann integral but it is used for all axiomatic definitions of integrals.

The partial derivatives, $\partial y / \partial x_{i}$, requires a new symbol because in this case, we work with many independent variables $x_i$ and when we want to determine $dy$, i.e. how much it changes, we need to say not only how one of the $x_i$ variables changes, but how all of them change. So the partial derivative symbol $\partial$ is meant to be the same thing as $d$ but it also conveys the message that all the other variables $x_j$ instead of the actual $x_i$ that appears after $\partial$ in the denominator are kept constant.

For example, if you have $y=\sin(x_1)\exp(x_2)$, then $$\frac{\partial y }{ \partial x_1} = \cos(x_1) \exp(x_2)$$ You couldn't just write $dy/dx_1$ on the left hand side because the "identity" wouldn't hold in general. It would only hold assuming that $x_2$ is constant, i.e. $dx_2=0$. More generally, we would have $$dy = dx_1 \cos(x_1) \exp(x_2) + dx_2 \sin(x_1)\exp(x_2).$$ Note that you would have to erase the second term proportional to $dx_2$ if you wanted to write a simple expression for $dy/dx_1$: that's why $dy/dx_1$ is simply not equal to $\partial y / \partial x_1$ if there are many variables. So the special symbol for partial derivatives had to be introduced.

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(comment deleted) –  mike4ty4 Aug 1 '12 at 6:32