Proof of Area Invariance of Closed Curve

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Discussion Overview

The discussion revolves around the proof of the area invariance of a closed curve, specifically whether the area remains constant regardless of how the curve is divided into non-overlapping rectangles of varying orientations. The scope includes theoretical aspects of geometry and calculus.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant seeks to prove that the area of a closed curve is invariant under different splicing methods into rectangles.
  • Another participant questions the terminology used, suggesting that "spliced into" may be better expressed as "sliced into" non-overlapping regions and asks for clarification on how area is defined.
  • A participant defines the area of a rectangle as length times breadth and proposes that the area of a closed curve can be defined as the sum of the areas of non-overlapping rectangles as the number of rectangles approaches infinity.
  • One participant challenges the proof, arguing that the rectangles cannot be true rectangles and emphasizes the need for a limit as the width approaches zero, suggesting that integration is necessary without requiring proof.
  • Another participant points out the need for precision in defining the area of a region, questioning whether the approach of using rectangles can accurately approximate the area as the number of rectangles increases.
  • One participant acknowledges that even if the approach works, it would only demonstrate that the sum of areas of rectangles tends to the same limit regardless of orientation, and suggests that understanding measure theory is essential for correctly defining area.
  • A later reply affirms the intent to prove the invariance of area, indicating a lack of existing proofs on the matter.

Areas of Agreement / Disagreement

Participants express differing views on the validity of the proposed proof and the definitions used. There is no consensus on the correctness of the proof or the definitions of area, and multiple competing perspectives remain unresolved.

Contextual Notes

Participants highlight limitations in the definitions and assumptions regarding the area of a closed curve and the method of using rectangles, indicating that the discussion may depend on specific interpretations of area and integration.

anantchowdhary
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Hello!

Quite some time ago I'd asked for help with a proof that proves that area of a closed curve is invariant i.e : its independent of the way it is spliced into.

Say we splice a closed curve into one set of rectangles with parallel sides and we then splice an identical curve with rectangles with some different orientation, I basically sought to prove that area calculated by summing up areas in both cases would be equal.


Here i present a proof

I'd be grateful to members who could comment on the proof and check it for validity.
Thanks :D
 

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Please Help! 127 Views and not a single reply! :S

Kindly Help!
 
I have no idea what you mean by "spliced into". "Sliced into"? Do you mean "divided into non-overlapping subregions"? Also, how are you defining the "area" of a plane region?
 
Thanks a Lot for the reply! by splice i mean ''slice'' into non overlapping regions.

Firstly defining the area of a rectangle as its length*breadth, and then for any general closed curve defining its area as the sum of areas of 'n',non overlapping rectangles that it can be divided into.Where n--> infinity
 
Your proof fails because you can never get all the rectangles to be true rectangles so the formula lnwn + ln+1wn+1 + ... = A is not true. This is why you need a limit as w -> 0, and then you simply have integration which needs no proof.
 
@ mu naught : yes that is why i said 'n' rectangles where n-->infinity.In integral calculus nowhere do we prove that the area is invariant (i mean irrespective of orientation of coordinate axes in the case of integration)
 
It's tricky, but you need to be more precise than saying your definition of the area of a region A is the sum of the areas of n non-overlapping rectangles in A as n tends to infinity. What if I give you 10 rectangles that roughly provide the shape of the outer boundary of A and and then keep dividing the inner rectangles? The number of rectangles approaches infinity, but the area never gets closer to the intuitive area.
 
Even if you do get this to work, all you will have proven is that the sum of the areas of some rectangles tends to the same limit as that of some differently oriented rectangles, as their number goes to infinity. If you're interested in defining area correctly and proving its properties, you need to learn about measure.
 
@werg22 : Yes! This is exactly what i intended to prove,for which i have never seen a proof.
 

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