# B Is this proper or it is abuse of notation?

#### fbs7

Hmm... I see.. you think the article in Wikipedia is lacking precision? He went on this line of thought:

"The differential of a single-variable function $f(x)$ is a two-variable function $df(x,\Delta x)$ defined as

$df(x,\Delta x) = f'(x)*\Delta x$"

Then he says "one or two arguments may be suppressed", that is, $df(x,\Delta x)=df(x)=df$. I understood he's saying the 3 forms are the same thing, ergo defined the same way.

Then he goes on saying "since $dx(x,\Delta x)=\Delta x$" it is "conventional" to write $dx=\Delta x$. Now, that "conventional" is confusing to me. I first interpreted what he said as this... you take a variable x, then build a function $X(x) = x$, so that now we can define a differential of that function as $dX(x,\Delta x)=X'*\Delta x=\Delta x$, and then the "convention" was that whenever one refers to "$dx$" he really means "$dX(x,\Delta x)=\Delta x$" - under a limit, of course.

But I get that this is not right, so I think I just don't get the nature of $dx$. How would you describe the nature of $dx$? That's a variable, a function, or an operator?

Or, better yet, using the language the Fresh_42 taught me... what is the domain of $dx$?

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#### Mark44

Mentor
Hmm... I see.. you think the article in Wikipedia is lacking precision? He went on this line of thought:

"The differential of a single-variable function $f(x)$ is a two-variable function $df(x,\Delta x)$ defined as

$df(x,\Delta x) = f'(x)*\Delta x$"

Then he says "one or two arguments may be suppressed", that is, $df(x,\Delta x)=df(x)=df$. I understood he's saying the 3 forms are the same thing, ergo defined the same way.

Then he goes on saying "since $dx(x,\Delta x)=\Delta x$" it is "conventional" to write $dx=\Delta x$. Now, this part is confusing to me, maybe he meant that "conventional" as meaning under limit, like $lim_{\Delta x->0} dx = lim_{\Delta x->0} \Delta x$?
This is pretty unusual, in my experience, to say that $dx = \Delta x$. $\Delta x$ usually refers to a relatively small real number, while dx is an "infinitesimally small, but nonzero" number.
fbs7 said:
Maybe I just don't get the nature of $dx$; I just understood the nature of $df(x)$ as being a function, so I thought $dx$ would be another function. How would you describe the nature of $dx$? That's a variable, a function, or an operator?
It's a variable. The term "operator" can have different meanings, depending on context. Relative to calculus and differential equations, an operator applies some operation to a function and produces another function. For example, the $\frac d {dx}$ operator is applied to a differentiable function and produces its derivative.

#### fbs7

I see!! So $df(x,\Delta x)$ is a function, while $dx$ is a variable! I guess the domain of that variable is a very small region around zero! I'm going to ignore that thing in Wikipedia about $dx=\Delta x$, that does sound confusing to me.

So, if $x$ is a free variable, $x ∈ ℝ$ then $dx$ is also a free variable, $dx ∈ ℝ-${0}, and does not depend on $x$, but $dx$ needs to be very small (no idea how to write that correctly other than $dx ∈ ReallyReallyTinyℝ-${0}

On the other hand if $y$ is not a free variable, but is a function of $x$, so while we write $dy$ in the same way as $dx$, the meaning is completely different, for that $dy$ is actually a function $dy(x,\Delta x)$, so its value depends on x. I think I'm getting it!!! The traps of writing $dx, dy, dz$ all the same way!!

By the way, I'm still flabbergasted that $df(f,\Delta x)$ is a function - it makes such sense, and it's such a joy when things just fall in place!

"Is this proper or it is abuse of notation?"

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