WebFeb 2, 2016 · I think of covariant and contravariant vectors the way mathematicians do: A covariant vector describes a tangent direction at a point p of a space: It is the velocity vector of a certain curve (as well as many others), thought of as the trajectory of a moving point. The set of all tangent vectors at the point p is called the tangent space (of ... http://astro.dur.ac.uk/~done/gr/l4.pdf

Deriving the Covariant Derivative of the Metric Tensor

WebJun 29, 2024 · For this derivation, we first need to calculate the partial derivative of the covarinat metric tensor (which can be expressed, as the dot product of two covariant basis vectors). ∂ ω g μ ν = ∂ ω φ μ, φ ν = ∂ ω φ μ, φ ν + φ μ, ∂ ω φ ν . By the definition of the covariant derivative, acting on a vector field: ∇ ω F ... Webthe covariant basis vectors are still identified as but now are functions of position. The effect of nonconstant basis vectors is most evident when applying derivatives to vector and scalar fields. In Chapter 14, the gradient operation was discussed for skewed coordinate systems, where the basis vectors gradient was defined as problems with authority

Appendix F: Christoffel Symbols and Covariant Derivatives

WebA vector quantity considered to be invariant in space can be measured by a set of chosen basis vectors. ::)There two ways to describe the vector quantity in terms of the chosen basis vectors. ... Covariant/contravariant imply how the components of the vector change when the basis vector changes it's length. ::) Multiplication of two vectors in ... WebJan 12, 2024 · The intuition here is as follows: we define the dual basis to "correct for" all the departures from orthonormality of the original basis. So if the angle between two basis vectors in the original basis was acute, the angle in the new basis will be obtuse; if one if the basis vectors was longer in the original basis, it will be shorter in the new basis. A covariant vector or cotangent vector (often abbreviated as covector) has components that co-vary with a change of basis. That is, the components must be transformed by the same matrix as the change of basis matrix. The components of covectors (as opposed to those of vectors) are said to be covariant. See more 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 See more The general formulation of covariance and contravariance refer to how the components of a coordinate vector transform under a change of basis (passive transformation). … See more In a finite-dimensional vector space V over a field K with a symmetric bilinear form g : V × V → K (which may be referred to as the metric tensor), there is little distinction between covariant … See more The distinction between covariance and contravariance is particularly important for computations with tensors, which often have mixed variance. This means that they have both covariant and contravariant components, or both vector and covector components. The … See more In physics, a vector typically arises as the outcome of a measurement or series of measurements, and is represented as a list (or tuple) of numbers such as $${\displaystyle (v_{1},v_{2},v_{3}).}$$ The numbers in the list depend on the choice of See more The choice of basis f on the vector space V defines uniquely a set of coordinate functions on V, by means of See more In the field of physics, the adjective covariant is often used informally as a synonym for invariant. For example, the Schrödinger equation does not keep its written form under … See more problems with authority disorder