The second is the structure of the composition of jets. If are a pair of real-valued functions, then we can define the product of their jets via Here we have suppressed the indeterminate z, since it is understood that jets are formal polynomials. This product is just the product of ordinary polynomials in z, modulo . In other words, it is multiplication in the ring , where is the ideal generated by polynomials homogeneous of order ≥ k + 1. We now move to the composition of jets. To avoid unnecessary technicalities, we consider jets of functions that map the origin to the origin. If and with f(0) = 0 and g(0) = 0, then . The composition of jets is defined by It is readily verified, using the chain rule, that this constitutes an associative noncommutative operation on the space of jets at the origin. In fact, the composition of k-jets is nothing more than the composition of polynomials modulo the ideal of polynomials homogeneous of order . Examples: In one dimension, let and . Then and Jets at a point in Euclidean space: rigorous definitions Analytic definition The following definition uses ideas from mathematical analysis to define jets and jet spaces. It can be generalized to smooth functions between Banach spaces, analytic functions between real or complex domains, to p-adic analysis, and to other areas of analysis. Let be the vector space of smooth functions . Let k be a non-negative integer, and let p be a point of . We define an equivalence relation on this space by declaring that two functions f and g are equivalent to order k if f and g have the same value at p, and all of their partial derivatives agree at p up to (and including) their k-th-order derivatives. In short, iff to k-th order. The k-th-order jet space of at p is defined to be the set of equivalence classes of , and is denoted by . The k-th-order jet at p of a smooth function is defined to be the equivalence class of f in . Algebraic-geometric definition The following definition uses ideas from algebraic geometry and commutative algebra to establish the notion of a jet and a jet space. Although this definition is not particularly suited for use in algebraic geometry per se, since it is cast in the smooth category, it can easily be tailored to