I think you are doing a great job here. I looked through your sections on determinants and inverses, and it is so much better than the way I was taught: "this is the formula for inverting a 2x2 matrix. This bit is called the determinant." For me, learning always works best when I know what we are working towards and seeing how it works, and putting the formal definition first is rarely effective in that. Start with a specific example, concrete where possible, and generalize from that.
Indeed. I think one of the thing I wanted to focus on as well in this series was explaining exactly where a lot of the formulas came from. It was a great learning experience, since a lot of what I was taught was basically as you said "here is the formula for the determinant and oh look, NxN determinants can be worked out in such and such a way if you just apply this algorithm". It was super useful to decompose why that algorithm works from a visual perspective. Same thing with surface integrals.
There's no perfect order for everyone. The trick is to spiral around revisiting levels and styles (and don't worry if you don't grok a part yet) until they all start to gel and your brain forms aweb of connections.
So many people think the second book they read on a topic is soooo much clearer than the first. But it doesn't much matter which book they dead first vs second :-)
The 'second book effect' is an interesting point, but that phenomenon, by itself, would suggest that there are improvements to be made in how we teach things from the beginning.
