A geometric algebra (Clifford algebra) is an extension of elementary algebra to work with geometrical objects such as vectors. It is built out of two fundamental operations: addition and geometric product. The multiplication of vectors alone results in objects called multivectors, among which are bivectors, the name applied in this paper to the objects of the bivector field ℝ², corresponding to the field of vectors V₂. Compared with other formalisms for manipulating geometric objects, geometric algebra supports dividing by a vector. On the other hand, although rarely used explicitly, a geometric representation of complex numbers is implicitly based on its structure of the Euclidean 2-dimensional vector space. 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, integral identities for scalar and vector fields in V₂ are presented, which are vector analogues of the well-known integral identities of complex analysis. Consequtly, these undoubtedly completely new results open up a wide range of applications in all fields where real vector analysis is used, such as quantum physics, classical physics, and so on. Therefore, special attention is paid to the vector analogue of Cauchy's calculus of residues in the field V₂. Finally, at the very end, the algebraic structure of the three-dimensional vector field V₃ is presented, as well as the corresponding fundamental integral identities.