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.
MATH 241 and (MATH 240 or MATH 461); or MATH 340 and MATH 341
Curves in the plane and Euclidean space
Curvature and torsion, moving frames
Smooth surfaces in Euclidean space
Tangent spaces and normal vector fields
First and second fundamental form
Intrinsic geometry of surfaces