I think a functional (or CS) perspective is an interesting approach, but I think you can lose some nice results in introductory differential geometry by following just this course. For example, the Gauss-Bonet theorem doesn't appear to be covered, which is an incredibly beautiful result linking the geometry and topology of manifolds.
For a more classical introduction to differential geometry requiring only multivariate calculus and some real analysis/point set topology, Do Carmo's "Differential Geometry of Curves and Surfaces" is a great textbook.
I put together a summary (key definitions/theorems) from an undergraduate course following Do Carmo at [2].
Lee's books are great as well, very different perspective, audience, motivation and development, but great. I am working through his first on topological manifolds now.
Also, at about the same level of difficulty but a different perspective:
For a more classical introduction to differential geometry requiring only multivariate calculus and some real analysis/point set topology, Do Carmo's "Differential Geometry of Curves and Surfaces" is a great textbook.
I put together a summary (key definitions/theorems) from an undergraduate course following Do Carmo at [2].
[1]: http://www.amazon.com/Differential-Geometry-Curves-Surfaces-...
[2]: http://ajtulloch.github.io/PDFs/MATH3968LectureNotes.pdf