Sentence examples for by the usual gradient from inspiring English sources

In the Euclidean setting, with ({nabla _{mathbb {H}^n}}) replaced by the usual gradient and (rho ) by the Euclidean distance, such inequalities have a long history and wide range of applications (see, for instance, [4, 8, 13]).

For each (tinmathbb{R}), we denote by ((cdot,cdot)_{t}) and (vert cdot vert _{t}) the usual inner product and related norm in (L^{2}(mathcal {O}_{t})) and by (((cdot,cdot))_{t}) and (|cdot|_{t}) the usual gradient inner product and associated norm in (H^{1}_{0}(mathcal {O}_{t})).

In case where (L1,…,Ln) is the usual gradient field on RN, we recover the classical result for the divergence equation proved by T. De Pauw and W. Pfeffer.

Evolution of anisotropy is accounted for by introducing substructural deformation gradients which are linear maps similar to the usual deformation gradient.

The mathematical model allows for the dual mode of diffusion into the particle, with surface diffusion being driven by a chemical potential rather than the usual concentration gradient.

However, the level of reinforcement is not properly predicted by the usual mechanical models and it is qualitatively accounted for by assuming a composition gradient (interphase) between the hard domains and the matrix.

These have been succeeded by the usual quotidian models.

Of course by the usual modification, Δ ∞ = L ∞.

These estimates were obtained by the usual modulus of continuity.

They were cared for by the usual staff and at the usual times.

The cows were cared for by the usual staff.