I Changing Summation to Integral

This is the text from Reif Statistical mechanics. In the screenshot he changes the summation to integral(Eq. 1.5.17) by saying that they are approximately continuous values. However,I don't see how. Can anyone justify this change?
 

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A.T.

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This is the text from Reif Statistical mechanics. In the screenshot he changes the summation to integral(Eq. 1.5.17) by saying that they are approximately continuous values. However,I don't see how. Can anyone justify this change?
Do you understand the relation between summation and integrals?

 

sophiecentaur

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Can anyone justify this change?
Basic ideas that you start with in Calculus take you from big steps to small steps and then you look at the limit as the step size approaches zero. That sort of relationship can be regarded as as 'continuous'. BUT that doesn't apply to all relationships. Not all relationships or mathematical functions are 'continuous and differentiable' over their whole range and you cannot do simple calculus in those cases. It is lucky (?) that most of the Physics we start off with is amenable to basic calculus methods (differentiation and integration). If you want to get deep into mathematical analysis methods then it will make you able to make choices about when you can and when you can't use basic calculus but, if you are like most of us, you just stick to the 'rules' that they use in book work and you won't go far wrong.
 

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