Covariance and contravariance of vectors - Wikipedia
In physics, especially in multilinear algebra and tensor analysis, covariance and contravariance describe how the quantitative description of certain geometric or physical entities changes with a change of basis.
Covariant Tensor -- from Wolfram MathWorld
4 天之前 · A covariant tensor, denoted with a lowered index (e.g., a_mu) is a tensor having specific transformation properties. In general, these transformation properties differ from those of a contravariant tensor.
协变张量 - 百度百科
协变张量(covariant tensor)是指所有指标都是协变指标的张量。既有协变又有逆变的张量为混合张量。
The rules for transformation of tensors of arbitrary rank are a generalization of the rules for vector transformation. For example, for a tensor of contravariant rank 2 and covariant rank 1: T0 = @x 0 @x @x @x @xˆ @x0 T ˆ where the prime symbol identi es the new coordinates and the transformed tensor. 3.2. Metric tensor. The metric tensor de ...
直观理解Tensor calculus中的一些核心概念(4) - 知乎
一旦我们有covariant basis,就可以定义covariant metric tensor为点积的array. z_{ij} = \vec z_i \cdot \vec z_j. 点积的对称性意味着covariant metric tensor 也是对称的，因此有 z_{i,j} = z_{ji}, 下一步定义contravariant metric tensor为covariant metric tensor的矩阵的逆. (现在矩阵 …
Covariant transformation - Wikipedia
In physics, a covariant transformation is a rule that specifies how certain entities, such as vectors or tensors, change under a change of basis. [1] The transformation that describes the new basis vectors as a linear combination of the old basis vectors is defined as a covariant transformation .
2011年10月14日 · We are now ready to define the covariant derivative of a general tensor. Since we define covariant derivatives by reference to a local coordinate system where the covariant derivative becomes a partial derivative, it should satisfy a product rule. For example, if we take a tensor S of rank 2 2, we have S = Sαβ γδeα ⊗eβ ⊗ω γ ⊗ ...
Covariant tensor - Encyclopedia of Mathematics
2020年6月5日 · If $ s = 1 $, the covariant tensor is called a covariant vector; when $ s \geq 2 $ a covariant tensor corresponds in an invariant way with an $ s $- linear mapping from the direct product $ E ^ {s} = E \times \dots \times E $( $ s $ times) into $ K $ by taking the components of the covariant tensor $ t $ relative to the basis $ e _ {1} \dots e ...
Intuitive way to understand covariance and contravariance in Tensor …
Every quantity which under a coordinate transformation, transforms like the derivatives of a scalar is called a covariant tensor. Accordingly, a reasonable generalization is having a quantity which transforms like the product of the components of two contravariant tensors, that is
Defining a Covariant Tensor - Mathematics Stack Exchange
2023年10月2日 · We usually define covariant tensors as those which rotate in the same sense w.r.t the basis vector. It is often also stated that if $P$ is covariant, $P'_{\alpha}=\frac{\partial{x^{\beta}}}{\partial{x'^{\alpha}}} P_{\beta}$ .