#### Description

Starting with multi-variable calculus, this course will develop the theme of invariants attached to the intrinsic and extrinsic geometry of curves and surfaces. Using local coordinates, invariants will be defined, which will later turn out to be independent of the choice of coordinates. The contrasts between intrinsic and extrinsic concepts will be emphasized. The notion of a smooth submanifold will be explored in detail, as will various notions of curvature. The various notions of curvature of surfaces are related to curvature and torsion of curves. The contrast between local and global phenomena is also emphasized. In the past the course has dealt with surfaces of revolution, ruled surfaces, minimal surfaces, special curves on surfaces, Gauss's "Theorema Egregium" and the Gauss-Bonnet theorem.

#### Prerequisites

MATH 241 and (MATH 240 or MATH 461); or MATH 340 and MATH 341

#### Topics

Curves in the plane and Euclidean space

Curvature and torsion, moving frames

Smooth surfaces in Euclidean space

Tangent spaces and normal vector fields

Orientability

First and second fundamental form

Normal curvature

Intrinsic geometry of surfaces