'codimension'是英语单词,翻译为“共维数”,在数学领域中指两个子集的维数之差。
在拓扑学中,一个点在一个流形上的维数是该流形中包含该点的最大开球的维数。同样,一个子流形的维数被定义为它上面点的最大值。如果一个子流形的维数等于一个包含它的流形的维数,则该子流形的共维数为零。如果两个流形M和N中,N是M的子流形且M的维数为n,N的维数为m,则它们的共维数为n-m。
以下是9个含有"codimension"的例句:
1. The codimension of the intersection of two hyperplanes in R^n is 2.
(R^n中两个超平面相交的共维数为2。)
2. The codimension of the boundary of a manifold is one.
(流形的边界的共维数为一。)
3. The codimension of the singular locus of a holomorphic foliation F on a complex manifold X is at least two.
(复流形X上的全纯叶子F的奇异轨迹的共维数至少为二。)
4. The codimension of the critical set of a smooth function f: M→R is at least one.
(光滑函数f:M→R的临界集的共维数至少为一。)
5. The codimension of the set of critical values of a smooth function f:M→N is zero.
(光滑函数f:M→N的临界值的集合的共维数为零。)
6. The dimension of the set of singular points of a submanifold of a Riemannian manifold is at most half of the codimension of the submanifold.
(黎曼流形的子流形的奇点集的维数至多是子流形的共维数的一半。)
7. The codimension of the kernel of a linear transformation T:V→W is the rank of T.
(线性变换T:V→W的核的共维数是T的秩。)
8. The codimension of a point in a line is one.
(直线上一点的共维数为一。)
9. The codimension of a plane in three-dimensional space is one.
(三维空间中一个平面的共维数为一。)
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