Although not with physicists or engineers in mind, in Edinburgh we do have a third year (in a four-year degree) course on differential geometry focussing on surfaces embedded in $\mathbb{R}^3$ which uses nothing more than calculus (including several variable calculus and ordinary differential equations) and linear algebra. The course is modern in that it uses the language of differential forms (in $\mathbb{R}^n$, so no need of manifolds).

A list of the lectures in the course (at least the last time I taught it, which was in 2007-8) is the following:

1. Surfaces
2. Vector fields
3. One-forms and line integrals
4. Differential forms
5. Moving frames and connection forms
6. The fundamental forms
7. Curvature
8. The meaning of curvature
9. Isometry and Gauss’s Theorema Egregium
10. Geodesics
11. Integration
12. Minimal surfaces
13. Stokes’s Theorem
14. The Gauss–Bonnet Theorem

This is delivered in 16 50-minute slots. The course was designed by my colleague Toby Bailey and it is taken by 40-60 students every year, so it seems to be quite popular. I think it is a very good introduction to differential geometry and ending with Gauss–Bonnet gives a nice way to complete the course.