**[Cr:3, Lc:2, Tt:1, Lb:0]**

- Differentiation of vectors.
- Curves in the plane and in space, arc length, reparametrization.
- Curvature, torsion, Serret-Frenet formulae.
- Fundamental theorem of curves in plane and space.
- Surfaces in three dimension (2-manifolds), smooth surfaces.
- Tangents, normals, quadratic surfaces.
- Change of variable formula, surfaces of revolution.
- First and second fundamental forms, isometries, conformal mappings.
- Normal and principal curvatures, Gaussian curvature and the Gauss map.
- Geodesics, geodesic curvature, Gauss' theorema egregium.

Additional Topics

- Isoperimetric inequality, four vertex theorem.
- Area and volume integrals, surface area.

- L. Brand, Vector Analysis, Dover Publications (2006).
- A. Pressley, Elementary Differential Geometry, SUMS, Springer (2001).