Sentence examples for with respect to coordinates from inspiring English sources

Semicolon and comma, respectively, denote covariant differentiation with respect to g ij and partial derivatives with respect to coordinates.

The boundary value problems under consideration are solved by employing Fourier exponential transformation techniques with respect to coordinates directed along the interface line.

Modeling systems in quasi-coordinates may facilitate a subsequent controller design, e.g. for underactuated systems with passive wheels or when a control input is a composite quantity with respect to coordinates that describe motion.

The potential derivatives with respect to coordinates x, y, z are expressed in terms of the ones related to the local orbital triad {u, v, w} and then Txx, Tyy and Tzz are treated as the observables.

In phenomenological approach, one of the methods to describe the fractional nonlocal continua is based on the variational principle with Lagrangian density that contains fractional derivatives with respect to coordinates.

Since these require second derivatives of energy with respect to coordinates, a smaller basis set 6-31G(d) was used for the purpose (a calculation time of ~4 days on a 12-core processor is typical).

It is shown that the response predictions of the strain-compatible volume-averaging method are invariant with respect to coordinate transformations while those of the generalized method of cells are not.

Example 3.8 Let p, q be such that 1 ≤ p ≤ q < ∞ and A = ℓ p. The algebra A is a Banach sequence algebra with respect to coordinate-wise multiplication of sequences (see Example 4.1.42 of [15]).

With respect to coordinate gene expression, it is necessary to delineate if the stomach-specific genes and the housekeeping genes undergo concurrent over- and under-methylation changes.

However, like many multi-institutional organizations, the CTSC has a complex five-partner structure and so faces challenges with respect to coordinating the experts and resources located at different sites for collaborative research.

Since the laminate is made of several orthotropic layers with their material axes oriented arbitrarily with respect to laminate coordinates, the constitutive equations of each layer must be transformed to the laminate coordinates x, y, and z.