Classical What Calculus Book Can Help Understand Goldstein's Classical Mechanics?

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The discussion centers on understanding specific mathematical concepts in Goldstein's Classical Mechanics, particularly related to summation and integration. The original poster expresses confusion over the notation differences in physics compared to standard multivariable calculus, especially regarding time derivatives and summation indices. Responses clarify that the summation notation is used for systems with varying degrees of freedom and emphasize the importance of treating derivatives with respect to different variables as separate entities. Suggested resources for further study include Feynman's lectures on calculus of variations and a specific book on the subject, along with a free online resource that covers relevant chapters. These materials aim to help bridge the gap in understanding the calculus concepts applied in mechanics.
Shakir
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Hello PF

I have attached two screenshots from Goldstein's Classical Mechanics. Although I have done a course on multivariable calculus, I don't understand what is going on in this math part.

Could you please provide some online resources or suggest a book so I can understand this sort of calculus? I am really stuck here.
Screenshot_2017-02-01-18-18-59-245_cn.wps.moffice_eng.png
Screenshot_2017-02-01-18-21-15-694_cn.wps.moffice_eng.png
 
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Could you point out what part(s) of those two pages you have an issue with? at a glance, I don't see anything that wasn't covered in my Calculus II and III courses back in the 90s.

I learned calculus with Larson & Hostetler and Varburg & Purcell, by the way. I know there are better texts out there, though.
 
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Hello
My problem is with the summation and integration part. I do not know how to use this method. Besides I learned multivariable calculus from a MATH teacher. The notations used in physics are a bit different e.g representation of x dot x, double dot to express velocity, acceleration etc. It often becomes very confusing. Maybe I missed some concepts or did not learn properly.
 
Problem starts at 2.16 to the rest.
 
The only two pieces of notation I see that are not in a standard Calculus sequence are summing over I and the use of a dot instead of a prime for time derivative.

The sum over I just means to sum over all values of the index i. It's written that way so that the same equation can be used for systems with any number of degrees of freedom.

One thing that takes a bit of getting used to in mechanics is that you can take a derivative with respect to y-dot in an expression as though it is a totally separate variable to y. It feels odd, but it works that way and it's just one of those things that must be gotten used to.

Other than those two points, you might find the chapter on calculus of variations in Feynman's lectures helpful. It's available free here:

http://www.feynmanlectures.caltech.edu/II_19.html
 
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