A. Necessary conditions for extremum
Conversion of Eq. (6) to a standard form can be accomplished as follows. The extremum should be investigated with the squared vector function
Then the condition for the existence of an extremum should be defined on the basis of :
It is not difficult to see that the extremum of (A1) is the sum of the extrema of its two squared components.
The theory of the calculus of variations makes clear that the extremum of a nonstandard function of the type F(y(x)) = F
_{1}(y(x))F
_{2}
(y(x)) is to be found on the basis of the linear combination:
G(y(x)) = F
_{1}
(y(x)) + λ_{0}
F
_{2}
(y(x)), (A2)
where λ_{0} is to be obtained from the ratio:
and y
_{
extr
}
(x) is defined using the conditions for extremum.
The functionals in (A1):
can be represented as a product of two equal functions. Then, in the two cases of U
_{2}
(y(x)) and U
_{3}
(y(x)) a trivial value for λ_{0}; λ_{0} = 1 renders the problem for extremum of the sum of the squares of the functions to an extremum of the sum of their power of one and in a sense – to the sum of the extrema of their moduluses.
The integrals in (A1) should be transformed in parametric form with arguments t:
x = x
_{
1
}
(t); y = x
_{
2
}
(t)
Introducing also the condition for a constant perimeter, the calculus of variations problem is presented in a near standard form – the extremum of :
is to be found, having in view the limitation:
The necessary conditions for the existence of an extremum are defined using the Lagrange function:
where the Lagrange coefficient λ in this case is a constant, to be found according to the necessary conditions for extremum and the limitation (A5).
The necessary conditions for extremum can be defined by the Euler system of equations:
with boundary conditions:
It is known that the equations in this system are in linear relation. In this case, a calibration equation is recommended in the following form:
For this specific problem, the limitation condition (A5) can be taken as calibration equation.
Lagrange function (A6) leads to a system of differential equations in parametric form:
with boundary conditions (A9).
B. Sufficiency of the necessary conditions
Defining the conditions for sufficiency of the necessary conditions is a rather complicated procedure. Here, some general considerations can be noted.

1.
The obtained extremum function is a solution of the Euler equation system – in this case equations (10) and (11).

2.
If λ is considered a parameter of the extremum function family, for the accepted boundary conditions and varying λ, a central field of extremum functions is obtained. The numerical analysis can show that they do not cross, and that a conjugated point does not exist, therefore the Jacoby condition for sufficiency is met.
For the second partial derivatives
and , taking into account Eq (9), we obtain:
and
It is obvious that (B1) and (B2) can have positive and negative values, which allows the conditions of Legendre to be met:
; < 0
; > 0  a condition for minimum

4.
It can be verified, based on Eq. (9), that the third partial derivative:
exists, which is also a part of the sufficiency conditions.
C. Field induced by a system of excitation coils
A relatively arbitrarily shaped contour L is considered (Fig. 1) with a current i(t) that excites an induced electrical field
(x, y, z). The field is obtained from Eq. (4), or:
where
is a vector linear element of the integration contour (L) and r is the distance from the element to the point where
(x, y, z) is computed.
Difficulties also appear for cases of contours in a relatively general position with respect to the coordinate axes [13]. Such difficulties are not insurmountable, but should be dealt with depending on the specific problem in view.
The present investigation relates to fields induced by flat contours approximated by n linear segments (Fig. 3) located in the XOY and YOZ planes. Applying the calculus of variation approach, such an approximation is justified. The numerical solution (Fig. 2) suggests that any other type of approximation would involve computational difficulties without major impact on the practical results. The solution accuracy depends, of course, on the value of n. For the triangular and square shapes, n is defined and does not affect the accuracy.
For a system of k contours, the resulting induced field is obtained by superposition, based on the linear relation between
(x, y, z) and di / dt, as evident from Eq. C1. In this case, three coils (L
_{1}),(L
_{2}) and (L
_{3}) are involved, where (L
_{1}) and (L
_{2}) are symmetrically located in the XOY plane, and (L
_{3}) is in the YOZ plane, symmetrically oriented with respect to the negative Zaxis. The region to be stimulated is located in the positive halfspace defined by XOY and the positive Zaxis.
(x, y, z) is to be obtained by solving the contour integrals along (L
_{1}), (L
_{2}) and (L
_{3}):
After decomposition of
over the respective axes, the following relation is obtained:
For the contours approximated by n segments (Fig. 3), sums of integrals are obtained for (L
_{1}) and (L
_{2}) respectively:
and for (L
_{3}):
The segments l
_{
j
} are introduced in the contour integrals by the corresponding linear relations y = m
_{
jk
}
x + n
_{
jk
}, k = 1, 2 ; j = 1,2,...,n in the XOY plane and z = m
_{
j
}
y + n
_{
j
} in the YOZ plane.
The segments l
_{
j
} are introduced in the contour integrals by the corresponding linear relations y = m
_{
jk
}
x + n
_{
jk
}, k = 1, 2 ; j = 1,2,...,n in the XOY plane and z = m
_{
j
}
y + n
_{
j
} in the YOZ plane.