Tangents, normals, curvatures, grad, curl, Laplacian… Here is a nice primer on differential geometry published in the Reviews of Modern Physics. On the way, you might learn a bit of soft condensed matter too. According to the abstract, the necessary pre-requisite is just vector calculus:
This article presents an overview of the differential geometry of curves and surfaces using examples from soft matter as illustrations. The presentation requires a background only in vector calculus and is otherwise self-contained.
Here is a link to the preprint at arXiv. Nice and extremely readable article!
