I am a physicist trying to learn vector calculus in a way that is a mixture of the way mathematicians learn it with the way that physicist learn it in order to be able to learn Differential Geometry the right way afterwards.
After asking myself what the geometrical meaning of the chain rule is, I came to the conclusion that I haven't got the right intuition to figure it out on my own.

So, I wanted to ask if anybody knows of a book on Vector Calculus that deals with the subject material in such a geometrically intuitive manner.

Note: I don't want a book that only explains things qualitatively. I want a book like, say Marsden and Tromba's, but with more emphasis on intuition.

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    $\begingroup$ Instead of looking for just 1 book, why not actually buy a physics book to see how they learn it, and buy a decently rigorous math book so your knowledge is adequate for starting differential geometry? Many physics books give an introductory chapter on vector calculus, like Griffiths intro Electrodynamics (pdf), John Thornton's classical dynamics of particle's systems, and many more. However, I am not sure if these books will impress, because lacking rigor and proofs doesn't necessarily mean they provide a more geometric picture... many times they say, "these are the tools, go use them". $\endgroup$ – Merkh Jun 30 '16 at 11:01
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    $\begingroup$ Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach by Hubbard and Hubbard...? Separately, Solid Shape by Jan J. Koenderink might be of interest, though it's perhaps more qualitative than you're seeking. $\endgroup$ – Andrew D. Hwang Jun 30 '16 at 11:03
  • $\begingroup$ @Merkh that approach won't learn me vector calculus properly I think. It's like you said it, they just give you the tools. For example, there is no discussion for the tangent plane. The first time somebody sees something that advanced in physics is in a general relativity course, but I want to tackle that course while I already know vector calculus really well. $\endgroup$ – TheQuantumMan Jun 30 '16 at 11:10
  • $\begingroup$ @AndrewD.Hwang I will try out the Hubbards book, it seems good. I will also check out Solid Shape, thank you. $\endgroup$ – TheQuantumMan Jun 30 '16 at 11:10
  • $\begingroup$ You're welcome; hope the suggestions are helpful! If you can borrow either or both through a library, I think that's a good idea. The Hubbards' book is a textbook, but conversational and sometimes digressive (in a good way). Koenderink's style is idiosyncratic, in a way that probably one either likes or hates.... $\endgroup$ – Andrew D. Hwang Jun 30 '16 at 11:18

With respect to an example about the chain's rule consider that a sequence of maps $\Bbb R\stackrel{\alpha}\to\Bbb R^2\stackrel{\Phi}\to\Bbb R^3$ allows you to control the movement along a curve in a surface: The surface will be the image of $\Phi$, let's dubbed it $\Sigma$, and the curve $C=\Phi\circ\alpha$ in that surface.

Now, knowing that $C'$ is a tangent vector to the curve $C$ but also to the surface $\Sigma$, one find that the components of $C'$ in the tangent space $T_p\Sigma$, where $p=C(t_0)$, can be found as follows:

The derivatives are connected through \begin{eqnarray*} C'&=&(\Phi\circ\alpha)'\\ &=&J\Phi\cdot \alpha', \end{eqnarray*} and if the Jacobian $J\Phi$ has in its columns two triplets which allows you to generate the space $T_p\Sigma$, then \begin{eqnarray*} C'(t_0)&=&J\Phi_{C(t_0)}\cdot\alpha'(t_0)\\ &=&J\Phi_p(v'(t_0)e_1+w'(t_0)e_2)\\ &=&v'(t_0)J\Phi_pe_1+w'(t_0)J\Phi_pe_2\\ C'(t_0)&=&v'(t_0)\partial_1+w'(t_0)\partial_2 \end{eqnarray*} where you can see that, at the instant $t_0$, the components of the tangent to curve $\alpha$ in $\Bbb R^2$ are the same of $C'$ but in a different basis.


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