## Understanding Varieties

Varieties are a basic structure in algebraic geometry. They were the central objects of study before Grothendieck reinvented the entire theory in his treatise Éléments de géométrie algébrique by introducing schemes. In the following we will introduce varieties and define algebraic curves.

Tangent vectors of functions are discussed early on in a standard calculus course. They are described either as directional derivatives or as velocities as curves. In manifold theory, we would like to generalize these ideas of calculus on $\mathbb{R}$ to calculus on manifolds. While there is an algebraic and geometric viewpoint of tangent vectors on manifolds, the algebraic realization is often faster to devlope the theory with. However, the geometric realization can be incredibly useful for computations. In the following, we’ll discuss both the algebraic and geometric viewpoints of tangent vectors, prove the equivalence between then, and see how both are useful by discussing the differential of a map.