First, geometric structures such as that of a differentiable manifold or a scheme can be expressed in terms of a sheaf of rings on the space.

Linearization is an effective method for approximating the output of a function formula_1 at any formula_2 based on the value and slope of the function at formula_3, given that formula_4 is differentiable on formula_5 (or formula_6) and that formula_6 is close to formula_5.