B Is this proper or it is abuse of notation?

  • Thread starter fbs7
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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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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.
 
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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!
 

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