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

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In summary, the conversation discusses the difference between dx times dy and a point. The question is clarified to be asking about the difference between an infinitely small change in the x and y directions and a geometrical object. The conversation also delves into the concept of a point as an element of a vector space and the use of dx*dy in mathematical processes. Different definitions and conceptualizations of a point are also mentioned. Overall, the conversation highlights the distinction between dx*dy as an algebraic object and a point as a geometrical object.
  • #1
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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  • #2
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:
 
  • #3
No, I mean an infinitely small change in the x direction times an infinitely small change in the y direction.
 
  • #4
it would be an infinitely small change in the area enclosed by a curve(s) in a plane. a point is just a point.
 
  • #5
A point is a geometrical object. dx dy is an algebraic object.
 
  • #6
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?
 
  • #7
A point is an element of a particular vector space (or, roughly, number system), for example R2
 
  • #8
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.
 
  • #9
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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FAQ: What's the Difference Between dx*dy and a Point?

What is the concept of integration?

The concept of integration refers to the process of combining different parts or elements into a whole. In mathematics, it specifically refers to the process of finding the area under a curve by dividing it into smaller, simpler parts.

What is the difference between indefinite and definite integration?

Indefinite integration involves finding a general function whose derivative is a given function, while definite integration involves finding the numerical value of the area under a curve between two specific points.

What are the different methods of integration?

The most commonly used methods of integration are substitution, integration by parts, and partial fractions. Other methods include trigonometric substitution, integration using partial derivatives, and integration by trigonometric identities.

What is the purpose of integration in real-life applications?

Integration is used in various real-life applications such as calculating areas, volumes, and velocities, as well as in physics, engineering, and economics to model and solve complex problems.

What are the common mistakes to avoid in integration?

Some common mistakes to avoid in integration include forgetting to add the constant of integration, making incorrect substitutions, and not properly evaluating limits. It is also important to check for symmetry and use the correct method for integration.

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