Cauchy kernel is a fundamental solution of the Cauchy–Riemann equations. Note that for smooth complex-valued functions of compact support on the generalized Cauchy integral formula simplifies to and is a restatement of the fact that, considered as a distribution, is a fundamental solution of the Cauchy–Riemann operator . The generalized Cauchy integral formula can be deduced for any bounded open region with boundary from this result and the formula for the distributional derivative of the characteristic function of : where the distribution on the right hand side denotes contour integration along . Several variables In several complex variables, the Cauchy integral formula can be generalized to polydiscs . Let be the polydisc given as the Cartesian product of open discs : Suppose that is a holomorphic function in continuous on the closure of . Then where . In real algebras The Cauchy integral formula is generalizable to real vector spaces of two or more dimensions. The insight into this property comes from geometric algebra, where objects beyond scalars and vectors (such as planar bivectors and volumetric trivectors) are considered, and a proper generalization of Stokes' theorem. Geometric calculus defines a derivative operator under its geometric product — that is, for a -vector field , the derivative generally contains terms of grade and . For example, a vector field () generally has in its derivative a scalar part, the divergence (), and a bivector part, the curl (). This particular derivative operator has a Green's function: where is the surface area of a unit -ball in the space (that is, , the circumference of a circle with radius 1, and , the surface area of a sphere with radius 1). By definition of a Green's function, It is this useful property that can be used, in conjunction with the generalized Stokes theorem: where, for an -dimensional vector space, is an -vector and is an -vector. The function can, in principle, be composed of any combination of multivectors. The proof of Cauchy's integral theorem for higher dimensional spaces relies on the using the generalized Stokes theorem on the quantity and use of the product rule: When , is called a monogenic function, the generalization of holomorphic functions to higher-dimensional spaces — indeed, it can be shown that the Cauchy–Riemann condition is just the two-dimensional expression of the monogenic condition. When that condition is met, the second term in the right-hand integral vanishes,