B When do we use which notation for Delta and Differentiation?

[shintashi]

I was taking notes recently for delta y/ delta x and noticed there's more than one way to skin a cat... or is there?

I saw the leibniz
dy/dx,

the triangle of change i was taught to use for "difference"
Δy/Δx,

and the mirror six
∂f/∂x

which is some sort of partial differential or something. And then I was confused about the backwards 6 and the lower case greek d
δy/δx

Can someone please sort out these four deltas?

 

[fresh_42]

I was taking notes recently for delta y/ delta x and noticed there's more than one way to skin a cat... or is there?

I saw the leibniz
dy/dx,

the triangle of change i was taught to use for "difference"
Δy/Δx,

and the mirror six
∂f/∂x

which is some sort of partial differential or something. And then I was confused about the backwards 6 and the lower case greek d
δy/δx

Can someone please sort out these four deltas?

You are basically right.

$\dfrac{dy}{dx}=\dfrac{d}{dx}y$ is the differential of a function $y=y(x)$ in one variable $x$.

$\dfrac{\Delta y}{\Delta x}=\dfrac{y_2-y_1}{x_2-x_1}$ is the quotient of two differences.

In case we consider a derivative of $y(x)$ at $x=x_0$, we get $\left. \dfrac{d}{dx}\right|_{x_0}y(x)=y'(x_0)=\lim_{x \to x_0}\dfrac{y(x)-y(x_0)}{x-x_0}=\lim_{\Delta x \to 0}\dfrac{\Delta y}{\Delta x}$ where the last equation isn't very accurate, because it doesn't mention the point $x_0$ explicitly. So it's only used for general considerations or when it's clear which differences must be taken.

$\dfrac{\partial y}{\partial x}$ is the partial derivative along the $x-$coordinate. This is needed, if $y=y(x,z)$ depends on more than one variable, say $x$ and $z$. In this case, if we take the partial derivative along $x$, then $z$ is considered constant, and vice versa. The $\partial$ is thus used to denote the fact, that there are more directions than only $x$.

$\dfrac{\delta y}{\delta x}$ in this context is probably the same as $\dfrac{\partial y}{\partial x}$ and only chosen as a matter of taste by the author. Usually the Kronecker delta $\delta$, which it is called, is used in a different context.

In case two different notations for different objects are needed, e.g. partial derivatives as linear form or as vector, then it might be necessary to distinguish the two, but as far as I know, there is no rule for the use of the Kronecker delta as partial derivative.

 

[jedishrfu]

The classic dy/dx is used for simple functions dependent on x only.

The partial derivative notation is for function which depend on two or more independent variables. To evaluate you take the derivative off one keeping the other independent variables constant. The result gives you the slope in the direction of the independent variable.

The delta notation is used mostly in limit expressions where the delta x is made to approach zero and the delta y / delta x is evaluated to give the slope.

The Greek delta notation is Greek to me :-)

@fresh_42 will know, I think he understands Greek.

 

[Mark44]

You are basically right.
$\dfrac{dy}{dx}=\dfrac{d}{dx}y$ is the differential of a function $y=y(x)$ in one variable $x$.

The above is the derivative of y with respect to x.

If y = f(x), then dy = f'(x)dx is defined as the differential of y.

 

[kuruman]

I will attempt to explain the differences based on usage I have seen.

$\Delta y$ is used to signify a finite difference as opposed to $dy$ that signifies an "infinitesimally small" difference. The convention for $\Delta y$ is "subtract from the value of $y$ that occurs later the value of $y$ that occurs earlier." "Earlier" and "later" are used with the understanding that there is a path specified by an independent variable on which $y$ depends. The distinction between the $dy$ and $\Delta y$ is exemplified by the equation $\int_a^b dy = \Delta y.$ This is shorthand notation for the English statement "If you add infinitesimally small differences in $y$ starting at its value at point $a$ and ending at its value at point $b$, the result will be the overall difference $y_b-y_a$."

Assume the independent variable is time $t$. Then $dy/dt$ is the rate of change of $y$ with respect to $t$ at a specific value of $t$. The ratio $\Delta y/ \Delta t$ is the ratio of overall changes between two points. The two convey different ideas. The average velocity is $\bar v=\Delta y /\Delta t$ while the instantaneous velocity is $v = dy/dt.$ To find the former, one needs only to know the endpoints in position and time. To find the latter, one needs to know how $y$ depends on $t$, i.e. $y(t)$ and a specific time value between the end points at which the ratio is to be evaluated.

The use of $\delta y$ is less frequent and I agree that, to some extent, it is a matter of preference. I have seen it used as a "small, but not infinitesimally small" difference, in other words smaller than the overall difference $\Delta y$ and larger than $dy$. I have not seen $\delta y/\delta t$ as denoting a partial derivative.

As others have already noted, one uses partials in cases where one has two independent variables, say $y = y(x,t)$.

 

[jedishrfu]

I've seen some examples where the $\delta$ is used as a specific delta of x:

$x + 0.00001 = x + \delta$ where $\delta$ is a value that can be made arbitrarily small for computing steps in an algorithm.

so as to distinguish it from $\Delta x$ that is $\delta = \Delta x$

 

[kuruman]

I've seen some examples where the $\delta$ is used as a specific delta of x:

$x + 0.00001 = x + \delta$ where $\delta$ is a value that can be made arbitrarily small for computing steps in an algorithm.

so as to distinguish it from $\Delta x$ that is $\delta = \Delta x$

In that context, $\delta$ plays the same role as $\epsilon$, often as a small expansion parameter.

 

[jedishrfu]

In that context, $\delta$ plays the same role as $\epsilon$, often as a small expansion parameter.

Yes, I've seen both ways.

 

[fresh_42]

I think except of $\delta_{ij}$ the small letter d of the Greek alphabet isn't associated with a certain meaning. E.g. we have the $\varepsilon - \delta$ definition of continuity, where it is just another small number. I can also imagine, that older books which must have been typeset used $\delta$ instead of $\partial$ simply because, the Greek alphabet has been in use anyway, and special signs as the partial were not. O.k. that's a guess, but it wasn't always as easy as today to change a typeset.

 

[jedishrfu]

I think except of $\delta_{ij}$ the small letter d of the Greek alphabet isn't associated with a certain meaning. E.g. we have the $\varepsilon - \delta$ definition of continuity, where it is just another small number. I can also imagine, that older books which must have been typeset used $\delta$ instead of $\partial$ simply because, the Greek alphabet has been in use anyway, and special signs as the partial were not. O.k. that's a guess, but it wasn't always as easy as today to change a typeset.

That's a pretty good guess and I believe its right.

In the 1980's computer world of QWERTY keyboards, Iverson gave a talk on APL at my college. Iverson was the inventor of APL, a mathematical computer language where every important CPU data operation was represented by a greek letter or some overstrike of greek letters. At the conclusion of the talk, someone asked about how are we to use this language on our keyboard since our keyboards don't have the greek alphabet (IBM sold APL and had the only keyboard that supported APL).

Iverson stared the guy down and said you expect ME to design MY language around YOUR keyboard and the guy just melted into his seat.

**APL = A Programming Language

https://en.wikipedia.org/wiki/APL_(programming_language)

 

[fresh_42]

I even remember that we built our own signs pixel by pixel on an Atari ...

 

[Mark44]

I even remember that we built our own signs pixel by pixel on an Atari ...

You had pixels?
We had to set banks of switches on the front of the computer!

 

[jedishrfu]

You had pixels?
We had to set banks of switches on the front of the computer!

I had just potentiometers, meters, switches, christmas tree bulbs and batteries. My computer was a precursor to a precursor microcomputer that used paperclips and aluminum foil for reading holes etched in paper. I also had BIG dreams of joining Starfleet before Star Trek gotten taken off the air. Of course, my favorite episode was City on the Edge of Forever where Spock makes a computer using vacuum tubes and some jewelers toolkit.

Hopefully we've answered the OPs question and don't derail this thread with our musings on the past.