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I have seen some EM texts use this symbol: [tex]d^3 x[/tex]. This just means the same thing as [tex]dV[/tex], right? Also, are there any advantages of using [tex]d^3 x[/tex] other than that it confuses people? :tongue:
cesiumfrog said:The real advantage is that it's shorter than writing "dxdxdx" everytime you do what is actually a sequence of three successively nested integrals. Technically you should use three integrand signs at the front and at least a couple sets of brackets between to make the old notation completely unambiguous, but we don't always bother with that either.
Compared to dV, the "advantage" is less abstraction. Conceptually, dV conveniently (and potentially confusingly) let's you imagine doing "just one" integral over volume, but when you actually have to get a pen and explicitly solve it, what you actually must do is integrate over each of the *three* axes *in turn*.
leright said:but if you're integrating in cartesian you're not integrating wrt to dxdxdx, but dxdydz, so that doesn't make much sense. I don't like that notation at all.
What about the above question?Swapnil said:What if your volume is not in cartesian coordinate. Say you are working in spherical coordinates, would you still use the notation [tex]d^3x[/tex] for the infinitesimal volume or is there a separate notation for that?
No. One reason we have [itex]d^3 \vec{x}[/itex] is to remind you about the units. So you have a volume, and therefore under the integrand you need a per volume thing (i.e. a density) for it to make sense.Swapnil said:What about the above question?
nrqed said:well, it's a matter of taste. It's like [itex] \frac{dy}{dx}[/itex]. It does not make sense since a derivative is not a ratio of two quantities but a limit. Yet, it's accepted as a notation. I guess that one has to be a bit flexible with these things.
Regards
[tex]\frac{\partial{y}}{\partial{x}}\frac{\partial{z}}{\partial{y}}\frac{\partial{x}}{\partial{z}}=-1[/tex]leright said:I consider dy/dx very much a ratio. It is the limit of a ratio. ...
In physics this is the way to think of things intuitively.
But instead of [tex]d\mathbf{x}[/tex] wouldn't a better notation be [tex]d^3\vec{s}[/tex]? I say this for three reasons:ObsessiveMathsFreak said:Personally I like to use [tex]d\mathbf{x}[/tex], where [tex]\mathbf{x}[/tex] is understood to be "n" coordinates.
Analagously, you should really use [tex]dx^3[/tex] for integrals.
[tex]\iiint f(x) dx^3 \equiv \iiint f(x) (dx_i)^3 \equiv \iiint f(x) dx_1dx_2dx_3[/tex]
OK a lot of that is nonsense, but if you wrote
[tex]\iiint f(x) d^3x[/tex]
It seems to imply that you mean to integrate three times with respect to the same variable x, not with respect to three different variables. Anyway this is why I prefer [tex]d\mathbf{x}[/tex].
The notation "d^3x" is used to represent the third derivative of a function with respect to the variable x.
Using "d^3x" notation allows for a compact and concise representation of higher order derivatives, making it easier to write and manipulate mathematical expressions.
Scientists use "d^3x" notation in various fields of research, particularly in mathematical modeling and analysis of physical systems. It is also commonly used in fields such as engineering, physics, and economics.
Yes, "d^3x" notation can be extended to represent derivatives of any order. For example, "d^4x" would represent the fourth derivative and so on.
Geometrically, "d^3x" represents the third derivative as the rate of change of the slope of a curve. It can also be interpreted as the curvature of a curve.