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What is the meaning of the term "prolongation" in differential geometry? Differential geometers often talk about "prolonging" a system of differential equations, or jet prolongation of bundle sections, but I don't really understand what mental picture the term "prolongation" is supposed to convey. Is it because when you introduce new variables for higher derivatives in a differential equation the system becomes "longer" when you write it down? Is that all there is to it, or is there some better reason for the terminology?

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Actually not the system is becoming "longer" but the space of dependent variables. Basically, you want a geometric object not just representing the dependent and independent variables but also the appearing partial derivatives. So you "prolong" the space of dependent variables $U$ by spaces representing the partial derivatives of order $n$ denoted by $U_n$. Then $U^{(n)} = U \times U_1 \times \dots \times U_n$ is the $n$th prolonged space.

If you have a smooth function $u = f(x)$ with $f \colon X \to U$ then its $n$th prolongation is $u^{(n)} = \mathsf{pr}^{(n)} f$ given by the partial derivatives up to order $n$. For example the $2$-prolongation of a function $f(x,y)$ would be $(f, \frac{\partial f}{\partial x},\frac{\partial f}{\partial y},\frac{\partial^2 f}{\partial x^2}, \frac{\partial^2 f}{\partial x \partial y}, \frac{\partial^2 f}{\partial x^2})$, representing the Taylor polynomial of second order. So the term prolongation originates from the prolongation of the space of dependent variables.

See also the book of Olver: "Applications of Lie Groups to Differential Equations", Sec. 2.3 and 3.5.

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