Differential geometry is the study of geometry using the techniques of differential calculus, integral calculus, linear algebra, multi-linear algebra (though mostly differential calculus). The course MTH305A: Several variables calculus & differential Geometry is a basic course that introduces the fundamentals of differential geometry primarily by focussing on the theory of curves and surfaces in three space. Goal of this course is to study differential geometry of curves and surfaces in the Euclidean space of dimension three. For our purpose, we begin the course with studying several variable calculus. We study a few topics in several variable calculus, e.g., chain rule, inverse and implicit function theorem, Taylor's theorem and applications etc, those are essential to study differential geometry of curves and surfaces. Next, we study geometric properties of curves both local (e.g., tangent, normal, binormal, regularity, curvature, torsion etc.) and global (e.g., isoperimetric inequality). Finally, we study geometric properties of surfaces and their higher dimensional analogue to built a solid mathematical understanding of the fundamental notions of differential geometry an sufficient visual intuition of the subject.
