A book on Vector Calculus with emphasis on geometrical intuition 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.
Thanks!
 A: 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.
