Differential geometry studies smooth curves, surfaces, and manifolds by applying differential and integral calculus to their local (as opposed to global/topological) geometric properties, such as curvature, torsion, arc length, geodesics, and tangent planes. A central object is the implicit equation f(x,y) = C (a level set), which defines a curve as the locus of all points satisfying a scalar-valued condition, distinct from an explicit equation where one variable is solved directly in terms of the other; this set-notation formalism generalizes to curves in R^3 defined by intersections of surfaces (e.g., a cylinder and a plane condition). This framework underlies later differential-geometric constructions such as gradients, tangent and normal vectors, and the transition from implicit to parametric representations.
Differential geometry studies smooth curves, surfaces, and manifolds by applying differential and integral calculus to their local (as opposed to global/topological) geometric properties, such as cur…