What's the Difference Between dx*dy and a Point?

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SUMMARY

The discussion clarifies the distinction between "dx times dy" and a point in mathematical terms. "dx*dy" represents the differential of area, an algebraic object related to infinitesimal changes in a two-dimensional space, while a point is defined as a geometrical object with no dimensions. The conversation emphasizes that "dx*dy" cannot be visualized as a geometric entity but is understood through its application in calculus, particularly in Riemann sums and integrals. The definition of a point varies, with LeBesgue's conceptualization being highlighted as a clear interpretation, albeit requiring a certain level of mathematical understanding.

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CaptainJames
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What's the difference between dx times dy and a point? Having trouble thinking about this... it's been hurting my head, any help would be greatly appreciated.
 
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What is the context of the question? "dx times dy" might be the "differential of area". It is, in any case, a differential which doesn't have anything to do with a point.
The question seems to me to be a lot like asking "What is the difference between the number 5 and a point?":confused:
 
No, I mean an infinitely small change in the x direction times an infinitely small change in the y direction.
 
it would be an infinitely small change in the area enclosed by a curve(s) in a plane. a point is just a point.
 
A point is a geometrical object. dx dy is an algebraic object.
 
Hmm, could someone define a point for me... i think the problem is that I define a point as an indivisible amount of space, which is probably wrong, any help?
 
A point is an element of a particular vector space (or, roughly, number system), for example R2
 
A point is an abstraction invented by Euclid.

dx*dy cannot be "visualized" as if it were a geometric object. Instead, visualize what one does with it, in context. Doing 2d integrals? Visualize finite partitions of a region (perhaps a grid of widths ΔxΔy), their Riemann sums under a function f, and the limiting behavior of all that. dx*dy is an informal way of saying we're looking at some limiting behavior, of finite Δx*Δy. Infinitesimals don't exist on their own - they exist with reference to some limiting process we're describing. Thus dx*dy has more mathematical meaning then just the 'point' at which it is located.
 
I see my problem, I was trying to apply my world to mathematics too readily, thanks for the help everyone.
 
  • #10
I guess a point can be considered a geometric shape with n sides and area, perimeter and volume =0 like a triangle with one null angle...
 
  • #11
LeBesgue's definition of a "point" is, as far as I am concerned, the best
conceptualisation of this abstract object. The downside is that it requires a
certain mathematical background. He stills manages to make the difference between
the differential area dx*dy and a point quite clear though.
 
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