This section presents some key differential calculus concepts useful in algebraic geometry. It briefly skips many classical developments from differential geometry, such as connections, infinitesimal transformations, and jets, which can be naturally expressed in the context of schemes. It also omits special phenomena in characteristic p > 0, though some are studied in affine settings. For further details on differential formalism in preschemes, the reader is referred to Exposés II and VII of [42], as well as later chapters of this treatise. Section 16.1 discusses normal invariants of immersions. Given two locally ringed spaces (X, O_X) and (Y, O_Y), and a morphism f = (ψ, θ): Y → X, where the homomorphism θ#: ψ*(O_X) → O_Y is surjective, O_Y is identified with the quotient sheaf ψ*(O_X)/J_f. The sheaf ψ*(O_X) is then equipped with the J_f-adic filtration. The n-th normal invariant of f is defined as ψ*(O_X)/J_f^{n+1}, and the n-th infinitesimal neighborhood of Y for f is denoted Y_f^{(n)} or Y^{(n)}. The graded sheaf Gr_•(f) is associated with the filtered sheaf ψ*(O_X), and Gr_1(f) = J_f/J_f^2 is called the conormal sheaf of f. The sheaves O_Y(n) = ψ*(O_X)/J_f^{n+1} form a projective system of sheaves of rings on Y, with transition maps φ_{nm} identifying O_Y^{(n)} as the quotient of O_Y^{(m)} by the power (J_f/J_f^{n+1})^m of the augmentation ideal of O_Y^{(n)}. The spaces Y^{(n)} form an inductive system of locally ringed spaces, with canonical morphisms h_n: Y^{(n)} → X. The graded sheaf Gr_•(f) is a graded algebra sheaf over O_X = Gr_0(f), and the Gr_k(f) are O_Y-modules. An example shows that in general, the sheaves O_Y^{(n)} are not naturally O_Y-modules or O_Y-algebras. The existence of such a structure corresponds to a ring homomorphism λ_n: O_Y → O_Y^{(n)}, inverse to the augmentation map φ_{0n}. A proposition states that for a morphism f of preschemes, Gr(f) is a quasi-coherent graded O_Y-algebra.This section presents some key differential calculus concepts useful in algebraic geometry. It briefly skips many classical developments from differential geometry, such as connections, infinitesimal transformations, and jets, which can be naturally expressed in the context of schemes. It also omits special phenomena in characteristic p > 0, though some are studied in affine settings. For further details on differential formalism in preschemes, the reader is referred to Exposés II and VII of [42], as well as later chapters of this treatise. Section 16.1 discusses normal invariants of immersions. Given two locally ringed spaces (X, O_X) and (Y, O_Y), and a morphism f = (ψ, θ): Y → X, where the homomorphism θ#: ψ*(O_X) → O_Y is surjective, O_Y is identified with the quotient sheaf ψ*(O_X)/J_f. The sheaf ψ*(O_X) is then equipped with the J_f-adic filtration. The n-th normal invariant of f is defined as ψ*(O_X)/J_f^{n+1}, and the n-th infinitesimal neighborhood of Y for f is denoted Y_f^{(n)} or Y^{(n)}. The graded sheaf Gr_•(f) is associated with the filtered sheaf ψ*(O_X), and Gr_1(f) = J_f/J_f^2 is called the conormal sheaf of f. The sheaves O_Y(n) = ψ*(O_X)/J_f^{n+1} form a projective system of sheaves of rings on Y, with transition maps φ_{nm} identifying O_Y^{(n)} as the quotient of O_Y^{(m)} by the power (J_f/J_f^{n+1})^m of the augmentation ideal of O_Y^{(n)}. The spaces Y^{(n)} form an inductive system of locally ringed spaces, with canonical morphisms h_n: Y^{(n)} → X. The graded sheaf Gr_•(f) is a graded algebra sheaf over O_X = Gr_0(f), and the Gr_k(f) are O_Y-modules. An example shows that in general, the sheaves O_Y^{(n)} are not naturally O_Y-modules or O_Y-algebras. The existence of such a structure corresponds to a ring homomorphism λ_n: O_Y → O_Y^{(n)}, inverse to the augmentation map φ_{0n}. A proposition states that for a morphism f of preschemes, Gr(f) is a quasi-coherent graded O_Y-algebra.