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So assume that there is a scalar field in the shape of a circle. Now if we define the total intensity of the field we will have to integrate in polar coordinates the formula defining the intensity of the field across the whole circle. Say a formula of intensity is represented by some arbitrary expression $I(x,y)$ and total intensity $\int{I(x,y)}$

Now an approximate solution will be to perform either uniform random sampling and summing up the intensity values to determine approximate total intensity. In the following image a grid points are spread across finite interval and then summed up to get approximate total intensity of the circular field. $\Sigma{I(x,y)}$

Question: Can I say "Without loss of generality, a field is comprised of infinite points, a finite subset of which can be used to approximate the total intensity of a given field"?

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GENIVI-LEARNER
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No, this is not appropriate. The discrete and continuous problems are not equivalent. One is just an approximation of the other, and to justify the method you should substantiate why the approximation is acceptably small.

You might say "without significant loss of accuracy" (if this is true).

  • Alright, but i thought discrete is a subset of continuous, so say we have a real number line that is infinite, I can simply take few numbers out of that infinite numbers and call them finite set of discrete points.. Maybe this might be wrong understanding? – GENIVI-LEARNER Sep 22 '20 at 14:02
  • @GENIVI-LEARNER: why on Earth would you call that WLOG ? –  Sep 22 '20 at 14:03
  • "general" in a sense that i am extracting a subset from an infinte set inorder to approximate. So a concrete example of random sampling is provided. – GENIVI-LEARNER Sep 22 '20 at 14:06
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    @GENIVI-LEARNER: then you completely miss the meaning of WLOG. –  Sep 22 '20 at 14:07
  • yes i believe it wasnt correct as well. You mentioned One is just an approximation of the other, and to justify the method you should substantiate why the approximation is acceptably small did you mean "why the approximation is acceptibly close to the actual"? or "why the approximation error is acceptibly small"? – GENIVI-LEARNER Sep 22 '20 at 14:12
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    @GENIVI-LEARNER: both. –  Sep 22 '20 at 14:31