Recommended book: Sanjoy Mahajan, "Street-Fighting Mathematics:
The Art of Educated Guessing and Opportunistic Problem Solving" (2010)
The whole book can be downloaded, free for all.
I like almost everything about this book except the title. I think of
street fighting as nasty, dirty, and destructive, and this book is the
opposite of that.
It contains many good lessons on how to think like a physicist. Some
of the examples are purely mathematical, such has how to calculate π
to a million places (hint: use the arithmetic-geometric mean). OTOH a
great many of the examples are drawn from real-world physics. From the
foreword by Carver Mead:
> Street-Fighting Mathematics is a breath of fresh air. Sanjoy Mahajan
> teaches us, in the most friendly way, tools that work in the real
> world. Just when we think that a topic is obvious, he brings us up
> to another level. My personal favorite is the approach to the
> Navier–Stokes equations: so nasty that I would never even attempt a
> solution. But he leads us through one, gleaning gems of insight
> along the way.
> In this little book are insights for every one of us. I have
> personally adopted several of the techniques that you will find
> here. I recommend it highly to every one of you.
This is very different from a typical math book. The equations are all
there, as you would expect, but in between there are lots of diagrams
and words motivating the equations and explaining the ramifications.
Level: The corresponding MIT course 18.086 lists calculus I, calculus
II, and differential equations as prerequisites.