Algebraic Structures of 2D and 3D Fields of Real Vectors

On the basis of the isomorphic algebraic structures of the field of complex numbers ℂ and the 2-dimensional Euclidean field of real vectors V₂, in terms of identical geometric products of elements, this paper brings integral identities for scalar and vector fields in V₂, which are vector analogues of the well-known integral identities of complex analysis. Consequently, in this paper, Theorem 1., which is a generalized fundamental theorem of integral calculus in the field V₂, is the vector analogue of the Cauchy theorem of complex analysis. Therefore, special attention is paid to the vector analogue of Cauchy's calculus of residues in the field V₂. Finally, at the very end of the paper, the algebraic structure of the 3D field of vectors V₃ is presented, as well as the corresponding fundamental integral identities.