Regular points of potential functions

A regular point of a function of n variables F(xj. x ) is defined as the point not being a critical point of this function. In other words, at a regular point, let it be for example the point x = 0, the function gradient (cf. definition (2.2)) does not vanish  

In the case of a function of one variable, V(x), it could be represented in the vicinity of a regular point in the form V(x) = x, see equation (2.7a). In the case of a regular critical point of several variables, the function V(xx ) can be simplified in a similar way, without changing its local character nearby this point. 

The definition of local equivalence of a function (2.12) suggests that a simpler form of a potential function with the same local properties may be sought by way of change of variables. 

In that case, let us perform the following change of variables x - x in a potential function satisfying the condition (2.27) 

The transformation x - x is an allowed change of variables when it does not alter the character of the investigated regular point x 0. This is the case when the Jacobian of transformation (2.28) does not vanish nearby the point x = 0 

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