In nD differential geometry, basic geometric structures and properties are described locally by differentiable functions and equations with indices that obey Einstein summation convention. Although symbolic manipulati...In nD differential geometry, basic geometric structures and properties are described locally by differentiable functions and equations with indices that obey Einstein summation convention. Although symbolic manipulation of such indexed functions is one of the oldest research topics in computer algebra, so far there exists no normal form reduction algorithm to judge whether two indexed polynomials involving indices of different coordinate systems are equal or not. It is a challenging task in computer algebra. In this paper, for a typical framework—the partial derivatives in coordinate transformation matrix involved are of order no more than two (such as local computations of ordinary curvatures and tor-sion), we put forward two algorithms, one on elimination of all redundant dummy indices of indexed polynomials, the other on normalization of such indexed polynomials, by which we can judge whether two indexed polynomials are equal or not. We implement the algorithms with Maple V.10 and use them to solve tensor verification problems in differential geometry, and to derive automatically the transformation rules of locally defined indexed functions under the change of local coordinates.展开更多
基金supported by Major State Basic Research Development Program of China (Grant No.2004CB318001)Leading Academic Disciplines Program of Shanghai (Grant No. S30501)
文摘In nD differential geometry, basic geometric structures and properties are described locally by differentiable functions and equations with indices that obey Einstein summation convention. Although symbolic manipulation of such indexed functions is one of the oldest research topics in computer algebra, so far there exists no normal form reduction algorithm to judge whether two indexed polynomials involving indices of different coordinate systems are equal or not. It is a challenging task in computer algebra. In this paper, for a typical framework—the partial derivatives in coordinate transformation matrix involved are of order no more than two (such as local computations of ordinary curvatures and tor-sion), we put forward two algorithms, one on elimination of all redundant dummy indices of indexed polynomials, the other on normalization of such indexed polynomials, by which we can judge whether two indexed polynomials are equal or not. We implement the algorithms with Maple V.10 and use them to solve tensor verification problems in differential geometry, and to derive automatically the transformation rules of locally defined indexed functions under the change of local coordinates.
