China
Africa
Explore chapters and articles related to this topic
Tensors In General Coordinates
Published in C. Young Eutiquio, Vector and Tensor Analysis, 2017
Actually, the terms covariant and contravariant have to do with the manner in which the components of the vector transform under a change of oblique coordinate systems. To see how the components transform, suppose x*i is another oblique coordinate system determined by a basis e*i (1 ≤ i ≤ 3). Let a*i and a*i denote the components of the vector A with respect to the bases e*i and e*i, respectively. Then we have
Gauge Theories
Published in K Grotz, H V Klapdor, S S Wilson, The Weak Interaction in Nuclear, Particle and Astrophysics, 2020
K Grotz, H V Klapdor, S S Wilson
The term covariant derivative is a term borrowed from general relativity, where it refers to a derivative in curvilinear space–time coordinates (see e.g. Misner et al (1973)). In order to compare adjacent vectors, the normal derivative ∂μ must be corrected by addition of a curvature-dependent correction term (see Figure 4.1).
Phase transitions for the geodesic flow of a rank one surface with nonpositive curvature
Published in Dynamical Systems, 2021
K. Burns, J. Buzzi, T. Fisher, N. Sawyer
In this setting, there are two continuous invariant subbundles and of . Each of these is one dimensional and orthogonal to the flow direction in the natural Sasaki metric. As we explain in the next section, corresponds to the orthogonal Jacobi fields whose length is nonincreasing for all time and to the orthogonal Jacobi fields whose length is nondecreasing for all time. A vector is said to be regular if . Otherwise, v is said to be singular. If v is singular, and there are nonzero covariantly constant orthogonal Jacobi fields along the geodesic that has initial tangent vector v. It is easily seen that v is singular if and only if for all t. We denote by the set of all regular vectors and by the set of all singular vectors. The set is obviously open and invariant while is closed and invariant.