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I am trying to learn some calculus 3 and I understand HOW to do the problems but I just don't understand WHY I'm doing what I'm doing. So does anyone have any good recommendations on books that are really down to earth, and explain the concepts in terms that humans can understand. Here are the topics that I want to understand:

A. Calculation of Geometric Quantities
   1) Surface Area
   2) Arc Length
   3) Curvature of Paths
      a) velocity
      b) speed
      c) acceleration
         i. tangential component
         ii. normal component
B. Line integrals of vector fields
   1) fundamental theorem of line integrals
   2) green's theorem
   3) finding the underlying scalar field for a conservative vector field
   4) direct calculation of a line integral
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For A. you might try Stewart's multivariable calculus. For things in B. I'm a big fan of "Div grad curl and all that" by Schey. –  icurays1 Nov 17 '12 at 19:23
    
You could also give "multivariable calculus" a search on youtube. MIT and Berkeley both have complete courses available. –  icurays1 Nov 17 '12 at 19:34
    
I strongly recommend that you study the book of George Simmons: Calculus with Analytic Geometry (1985, 1996 (2nd ed.)). –  Elias Nov 17 '12 at 19:36
    
I also recommend Div Grad Curl and All That. –  littleO Nov 17 '12 at 20:08
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You might try Vector Calculus, Linear Algebra and Differential Forms by Hubbard & Hubbard - it's a nice introductory book that gives a first taste of the unified picture provided by forms. –  Neal Nov 17 '12 at 20:36
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It is generally accepted that science provides a description of 'HOW' nature works, not "WHY" it works this way. Mathematics often provides 'higher' concepts which allow science to make these descriptions accurately. By 'higher', we mean 'generalized through the use of abstract reasoning'.

To understand questions of 'WHY' usually involves the development and application of even higher concepts such as motivation, purpose and intention which are beyond the scope even of mathematics.

If you want to understand "WHY" certain topics are taught in the mathematics curriculum, it might be useful to try and get a grasp on where some of these mathematical topics are used in practice. To this end, a book on Engineering Physics might be helpful, such as Serway's 'Physics for Scientists and Engineers'.

The topics in part A are really fundamental to physics and engineering mechanics (both statics and dynamics). For example, surface area is important in calculating tension in mechanical components, as well as stresses and strain of engineering materials (eg: how many bolts of what size do you need to hold up a bridge?).

Topics in part B are also fundamental to physics, especially hydrodynamics and electrodynamics. Although these are covered at a basic level in the previous text, however, a more specialized text on either topic would also be useful (eg: Hydrodynamics by Horace Lamb or Engineering Electrodynamics by William Hayt).

Feynman's Lectures on Physics (parts 1 & 2) may also offer you some insights into the subtle connection between physics and mathematics by giving some practical applications of these mathematical topics.

A fascinating discussion on the relationship between physics and mathematics can be found in Feynman's 'Character of Physical Law', which is also available as a series of six videos online at Microsoft Research under 'Project Tuva' (see http://research.microsoft.com/apps/tools/tuva/index.html). Specifically, refer to Lecture 2: 'The Relation of Mathematics and Physics'.

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