n维空间上一类对称区域的n重积分性质及其应用研究

A Study on n-Degree Integral Properties and Applications of a Class of Symmetric Region on the n-Dimensional Space

n重积分与定积分的概念在数量关系上的一致性使它们具有诸多类似的性质,单变量奇偶函数在对称区间上定积分的运算性质,能够推广到n维空间一类对称区域的n重积分。通过讨论空间对称点的坐标轮换,以及对称点从对称区域Ω_1到Ω_2映射变换的Jacobian行列式,性质推广得以严格证明。结论作为基础理论具有实际应用价值:简化n维球体的面积公式推导;巧用对称性提高工程计算效率;帮助人们更好地理解和讨论n维空间的数学问题,构建良好的数学思想方法与数学解题行为。

The concept of n-degree integral and definite integral has a number of similar properties because of the consistency of quantitative relation. The operation properties of integral for univariate odd and even functions in symmetric domains can be generalized to n-degree integral in symmetric domains of n-dimensional space. By discussing the coordinate transformation of space symmetric points,Jacobian determinant for the projective transformation of the symmetric points projected from symmetric domain Ω_1to Ω_2,the generalized propevties are strictly proved. The conclusion as a basic theory has meaningful applied value. The derivation of the formula for calculating surface area of n-dimensional sphere can be simplified. Using symmetry to improve the efficiency of engineering computation. All of these have strong positive influences on forming mathematical thinking and problem-solving skills.