Let a functional $f$ be given on a space $X$, and let $V$ be a parameter space. A variation of the argument $x_0\in X$ is an ordinary curve $x(t,v)$, $\alpha\leq t\leq\beta$, $\alpha\leq0$, $\beta\geq0$, $v\in V$, in $X$ which passes through $x_0$ in a certain neighbourhood defined by the restrictions that are in force. Let the value $t=0$ correspond to $x_0$. As $v$ runs through the set of all parameters, the variations run through a certain family of curves issuing from $x_0$. In finite-dimensional and infinite-dimensional analysis, beginning with Lagrange, it is usual to employ the directional variation where $V=X$ and $x(t,v)=x_0+tv$. In this case the vector $v$ is referred to as the variation. However, other classes of variations are employed in geometry, in variational calculus and, in particular, in the theory of optimal control; these include polygonal variations, needle-shaped or spiked variations and variations connected with sliding regimes , . The choice of the space of variations and the construction of the variations themselves are a very important element in obtaining necessary conditions for an extremum. See also Variation of a functional; Gâteaux derivative; Fréchet derivative; Functional derivative.