The expected value of the line integral of one-forms
The author analyses from the point of view of measure theory the problem of approximating a line integral using the quotient of Newton instead of the derivative. It is demonstrated that the expected value of a certain scheme of approximation of line integrals of closed one-form \(\beta\), with respect to the Wiener measure, converges to the integral over the interval of the parameter, of the expected value of the adjoint \(\delta\) of the exterior derivative, applied to the one-form. Another result is that if the denominator of the quotient of Newton is risen to a power smaller than \(1/2\), and if the line integral of a harmonic form is approximated by using this new quotient, then for almost all continuous lines, the inferior limit of this scheme of approximation is zero.
scientific article; zbMATH DE number 5126089