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I will appreciate greatly if somebody could explain me how the two terms on the left are combined to form a total derivative? The first term on the left changed by using r=r_2*sin(phi). But the book says that there two terms that are combined.

This is not a homework, but a self-study.

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    what do you get if you differentiate $\frac{d}{d\phi}[(rN_{\phi})\sin\phi]$ using the product rule? – peek-a-boo May 09 '21 at 12:28
  • If I am not mistaken we get: rsin(phi) + rN*cos(phi). What do you suggest next, then? –  May 09 '21 at 12:36
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    Nope, try again. My point is that if you differentiate whats on the second line, you get exactly what's on the first line. That's why they say "combine the two terms". – peek-a-boo May 09 '21 at 12:37
  • based on the what's written, no $N$ is a function of $\phi$. – peek-a-boo May 09 '21 at 12:38
  • I got it. Thanks a lot. Now it makes sense. I have never met this formulation before. As we speak, I would like to ask another question: Can you recommend a book that goes mainly on application of calculus and DEs? I.e. application for real life examples and engineering –  May 09 '21 at 12:43
  • You should really ask that in a separate question. I do not have experience with books like that specifically (when starting out calculus, I pretty much just learnt from Khan Academy, and MIT lectures on physics), so you'll get better answers from others. – peek-a-boo May 09 '21 at 12:48
  • That OK. Again, I appreciate your help.

    Best regards, Sergey

     –  May 09 '21 at 12:50

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