MTH305A - Several Variable Calculus & Defferential Geometry

 Differentiation: Definition and examples, Mean value inequality, Tangent planes to level sets of functions; Implicit mapping theorem, Inverse mapping theorem and applications; Taylor's theorem and applications. Curves: Definition and examples, Regular curves, Plane curves, Curvature of plane curves, Isoperimetric inequality for plane curves; Space curves, FrenetSerret formula for space curves; Local existence theorem curves. Surfaces: Definition and examples; Tangent planes, Maps between surfaces; First fundamental and second fundamental forms; Curvature of surface; Hilbert's theorem for compact surfaces; Gauss theorem a Egregium.