Triangular integrals for 2-, 3- and 4-variable functions

Kiyohisa Tokunaga

doi:10.1186/1687-1847-2014-89

Abstract

The triangular integrals for 2-, 3- and 4-variable functions are respectively and precisely defined as the single limits of double, triple, and quadruple sums in detail. A corollary of the divergence theorem in each dimension is useful to determine the triangular integral value. The indices of the sequence of the integrand must coincide with those of the corresponding integral variable to calculate the correct triangular integral value. In a triangular triple integral, one kind of two sets of increments is inappropriate for the convergence of numerical values, but the other kind is able to calculate numerical values by a computer algebra system.

Highlights

IntroductionThe calculation process of the triangular quadruple integral for a -variable function in the D time-space is precisely defined as the single limit of quadruple dependent sums by
The primary theme of this article is a double integral for a -variable function p = p(x, y) in a domain D on the D plane
In addition to introducing triangular elements in the finite element method [ ], a combination of a triangular area method and double dependent series was applied to sweep all of the area [ ]

Summary

Introduction

The calculation process of the triangular quadruple integral for a -variable function in the D time-space is precisely defined as the single limit of quadruple dependent sums by This article includes revisions of the divergence theorems and the related corollaries based on the appropriate increments of the double and the triple sequences in the calculation processes of the triangular triple and quadruple integrals for - and -variable functions. 2.1 One kind of finite line element vector on the 2D plane For triangular double integral, the following increments of single sequence of points on the D plane are introduced. 3.1 Two kinds of finite area element vectors in the 3D space For a triangular triple integral, the following increments of the double sequence of points in the D space are introduced.

Triangular triple integral for a 3-variable function

Triangular quadruple integral for a 4-variable function