When we can change a sum to an integral?

In summary, in physics, a sum can be changed to an integral when the indexing value becomes continuous and the term of the sum is comparable to the sum. An integral is essentially the limit of a sum with an arbitrarily large number of arbitrarily small elements. However, if one or more terms of the sum are too large, it cannot be considered as an integral. This can be seen when using sums to approximate the area under a curve, where a smaller delta x and a more continuous x sequence leads to the use of an integral. In Bose-Einstein condensation, the number of particles in the ground state must be separated from the integral because it becomes very large and must be accounted for separately. Further discussions about this topic should be made in
  • #1
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In physics we often change a sum to an integral.But I am not clear when can we change a sum to an integral?When a term of sum is comparable to the sum,can we change the sum to integral?
 
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  • #2
When the indexing value i of the sum becomes continuous like x .
 
  • #3
An integral is essentially the limit of a sum an arbitrarily large number of arbitrarily small elements.

AM
 
  • #4
Then if one(or more) term of sum is very great(can not consider as a small element) so we can not consider as an integral?
 
  • #5
Think about how you initially use sums to approximate the area under a curve.

you might compute the area as the sum of box areas where the width of the box is one and the length of the box is f(x) and use the sum of boxes with
x=1, 2, 3, 4, 5...

then to get a better approximation you use a width of one half for the box and the sequence with x=0.5, 1, 1.5, 2, 2.5 ...

so as delta x (aka box width and x difference) gets smaller and the x sequence becomes more continuous then an integral comes into play.
 
  • #6
I do not understand why in Bose-Einstein condensation, below critical temprature we must separate the number of particles of ground state(E=0) from the integral?Is that because the number of particles in ground state become very large so that we must separate it from the integral(the total number of particles)?
 
  • #7
You should ask that question in a separate thread with a related title As now you’re talking more physics and not math.

closing this thread now.
 

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