协变和逆变向量
Covariant and Contravariant Vectors
摘要：
矢量是一个物理量，它不依赖于任何坐标系。它需要在某些基础上进行扩展，以便进行实际计算，并且它的组成部分确实取决于所选择的基础。在标准正交基中的展开在数学上是简单的。但在许多物理情况下，我们必须选择一个非正交基（或斜坐标系）。但向量在非正交基上的展开式不便于处理。利用向量的逆变分量和协变分量的概念，我们使非正交基具有标准正交基的性质。在量子力学中，Ket向量和Bra向量的概念是相同的，我们通过完备性关系比较了这两个等价的情形。这个概念出现在度量流形的微分几何中，其中考虑了基的非正交性以及非欧几里德几何。
英文摘要：
Vector is a physical quantity and it does not depend on any co-ordinate system. It need to be expanded in some basis for practical calculation and its components do depend on the chosen basis. The expansion in orthonormal basis is mathematically simple. But in many physical situations we have to choose an non-orthogonal basis (or oblique co-ordinate system). But the expansion of a vector in non-orthogonal basis is not convenient to work with. With the notion of contravariant and covariant components of a vector, we make non-orthogonal basis to behave like orthonormal basis. The same notion appears in quantum mechanics as Ket and Bra vectors and we compare the two equivalent situation via the completeness relation. This notion appears in the differential geometry of a metric manifold for tangent vectors at a point, where it takes into account the non-orthogonality of basis as well as non-Euclidean geometry.
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