Changing Summation to Integral

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• Abhishek11235
In summary, the conversation discusses a change from summation to integral in Reif Statistical mechanics, with one person questioning the justification for this change. The other person explains that this change is based on the idea of continuity, which applies to most mathematical relationships used in physics. However, there are cases where this may not be applicable, and advanced mathematical analysis may be needed.
Abhishek11235
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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Abhishek11235 said:
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?

Abhishek11235 said:
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.

1. What is the difference between summation and integration?

Summation is the process of adding together a finite number of terms, while integration is the process of finding the area under a curve. In other words, summation deals with discrete quantities while integration deals with continuous quantities.

2. Why is it necessary to change from summation to integration?

In many real-world situations, the data or function being analyzed is continuous and cannot be accurately represented by a finite number of discrete points. By changing from summation to integration, we can more accurately model and analyze these continuous quantities.

3. How do you change from summation to integration?

The process of changing from summation to integration involves taking the limit of a summation as the number of terms approaches infinity. This is known as the Riemann sum, and it can be represented using the integral symbol.

4. What are the applications of changing from summation to integration?

Changing from summation to integration is used in many areas of science and engineering, such as physics, chemistry, and economics. It allows us to accurately model and analyze continuous quantities, such as velocity, acceleration, and population growth.

5. Are there any limitations to changing from summation to integration?

While integration allows us to more accurately model continuous quantities, it does have its limitations. In some cases, the function being integrated may be too complex or the data may not be continuous, making it difficult to accurately use integration methods. Additionally, the process of changing from summation to integration may be time-consuming and require advanced mathematical knowledge.

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