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I was wondering if there is a book out there that doesn't teach you how to do calculus, but teaches you how to apply it in the physical or social sciences.

I know calculus, integration and differentiation and the applications for each, but I struggle to find a book that teaches me how to use calculus to derive my own mathematical models.

I think it will be easier to give an example:

Imagine a spinning rod with uniform mass with length, $L$. To find the kinetic energy of the rod I know that each part of the rod moves at a different speed, and therefore the entire kinetic energy of the rod is the sum of all the infinitesimals of little sections on the rod.

I am not going to solve this problem, because this isn't the question on hand, but the point is is that I am looking for a book that teaches you, and challenges you to use your already knowledge of calculus to solve problems like this?

Hope I am making myself clear, and any input would greatly be appreciated.

Uys of Spades
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    Excellent question. ${}\qquad{}$ – Michael Hardy Dec 01 '15 at 21:07
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    Doesn't every book on physics -- and in particular, classical mechanics -- "[challenge] you to use your already knowledge of calculus to solve problems like this"? –  Dec 01 '15 at 21:14
  • @Bye_World no it just teaches you how to use calculus to solve basic physics problems.Your own models you make don't come from books. I guess I am asking for a book that teaches you to think of the world in calculus and models instead of the basic physics question: You got $m,v,L$ find the kinetic energy of something, and the problem is already tailored and structured that you know you are going to use the concepts learned in a chapter, but with different values. – Uys of Spades Dec 01 '15 at 21:30
  • @UysofSpades Do you know how jazz musicians learn to improvise? Well, as one of them said, they might start from imitating the already existing music, then they assimilate it and after that they are ready to "innovate". You know, it's quite common process in other arts. And constructing mathematical models is a sort of an art, too. Frankly speaking, calculus to mathematical models is somewhat as sheet notation for music -- it's the language of writing, but not the writing itself. The understanding of research field might be more important than the ability to translate it to calculus or ODEs... – Evgeny Dec 02 '15 at 01:09
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    There are "general applied math" books, such as http://www.amazon.com/Introduction-Foundations-Applied-Mathematics-Texts-ebook/dp/B00DILHP9M/ref=sr_1_1?ie=UTF8&qid=1449018694&sr=8-1&keywords=applied+holmes which might be helpful. (This one is probably above your level if you haven't had at least elementary ODEs and linear algebra, though.) – Ian Dec 02 '15 at 01:12
  • @Evgeny that is exactly what I am looking for. Is something to teach me the art to give me problems and challenges that says you can solve this using calculus, and this is a way of looking at it. If calculus is the saxophone, then I would want something to teach me how to play it with 'style' – Uys of Spades Dec 02 '15 at 01:12
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    You might be interested in Mahajan's Street-Fighting Mathematics. I only know of it and haven't looked at it in any great detail, but it's free and worth looking at. – pjs36 Dec 02 '15 at 01:25
  • @UysofSpades:I'm not sure whether this book Calculus made easy:http://1drv.ms/1RvyMOk by Silvanus P.Thompson:https://en.wikipedia.org/wiki/Silvanus_P._Thompson would show you too much examples in physics but still it shows a few.It's definetly a intuitive book for Calculus.Also check my answer at:http://math.stackexchange.com/questions/1523296/developing-mathematic-intuition/1616696#1616696 – justin Jan 28 '16 at 08:43
  • @justin Thank you for that. I will definitely take a look – Uys of Spades Feb 03 '16 at 11:06

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