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Application of the Frobenius theorem to determination of potentials with their gradients orthogonal to vector fields

Kaczorek T.

Poznan University of Technology Academic Journals. Electrical Engineering

2007

No 56

7-12

The problem of finding potentials of the vector field those gradients are orthogonal to the given vector fields is formulated and solved. The solution to the problem is based on the Frobenius theorem. It is shown that the problem has a solution if and only if the distribution spanned on the vector fields is involutive. A procedure for finding of the desired potentials is proposed and demonstrated on an example of finding one potential for two given vector fields.

The Cauchy problem for certain generalized differential equations of second order

Poreda T., Szadkowska A.

Demonstratio Mathematica

2002

Vol. 35, nr 4

773-782

The present paper concerns some generalized differential equations of second order for mappings from a subset of Banach space into a Banach space. The subject matter refers to studies of generalized differential equations of the first order submitted in [4]. Let X,Y be Banach spaces over the field R and let U, V be open subsets of X and V, respectively. Let h1, h2 be mappings from U into X. In this paper we study the Cauchy problem D2f(x)(h2(x)) = F{x, Df(x)(h1(x))), x € U for mappings from a subset of a Banach space into a Banach space, which are defined in C class, at a neighbourhood of nonsingular point (that is, at a neighbourhood of such point xo for whichh1(xo)= 0 and h2(xo= 0).