In this book I present differential geometry and related mathematical topics with the help of examples from physics. It is well known that there is something strikingly mathematical about the physical universe as it is conceived of in the physical sciences. The convergence of physics with mathematics, especially differential geometry, topology and global analysis is even more pronounced in the newer quantum theories such as gauge field theory and string theory. The
amount of mathematical sophistication required for a good understanding of modern physics is astounding. On the other hand, the philosophy of this book is that mathematics itself is illuminated by physics and physical thinking.

The ideal of a truth that transcends all interpretation is perhaps unattainable. Even the two most impressively objective realities, the physical and the mathematical, are still only approachable through, and are ultimately inseparable from, our normative and linguistic background. And yet it is exactly the tendency of these two sciences to point beyond themselves to something transcendentally real that so inspires us. Whenever we interpret something real, whether physical or mathematical, there will be those aspects which arise as mere artifacts of our current descriptive scheme and those aspects that seem to be objective realities which are revealed equally well through any of a multitude of equivalent descriptive schemes-cognitive inertial frames as it were. This theme is played out even within geometry itself where a viewpoint or interpretive scheme translates to the notion of a coordinate system on a differentiable manifold.