Sentence examples for functional derivatives from inspiring English sources

We find that the naturally guessed equations of motion, as derived via partial functional derivatives from a free energy, are usually reasonable, only for materials with plastic effects this assumption is more delicate due to the presence of internal variables.

3 A number of concepts from the theory of functionals and their derivatives (see, e.g., Parr and Yang 1989; Davis 1996; Engel and Dreizler 2011 for more details) required for this purpose are briefly summarized in Appendix A: brief summary of concepts from the theory of functionals and illustrated in Appendix B: functional derivatives based on delta function.

A hybrid-adjoint Navier Stokes method for the pressure-based computation of hydrodynamic objective functional derivatives with respect to the shape is systematically derived in three steps: The underlying adjoint partial differential equations and boundary conditions for the frozen-turbulence Reynolds-averaged Navier Stokes equations are considered in the first step.

However, an algebraic treatment of the functional derivatives required in this report is sufficient.

This necessitates large-scale expansion of SCs followed by efficient and homogeneous differentiation into functional derivatives.

Elementary functionals and their functional derivatives are defined and calculated following the methods of Appendix A.1.

A closed equation for the one-point probability density function (PDF) for FSD has been derived, using the functional derivative technique, together with the assumption that the convection velocity and flame stretch are homogeneous, isotropic, Gaussian random fields with many spatial scales and rapid oscillations in time.

This δ-function-based form of the functional derivative is common in physics (e.g., Parr and Yang 1989; Davis 1996).

In the present model, the functional derivative of the nonlinear gain function is given by δ S [ u ( x, t ) ] / δ u ( x, t ) = δ ( u ( x, t ) − θ ).

Consequently, the functional derivative of the smallest eigenfrequency with respect to PCLS function takes nonzero value in the original material region and zero in the void region.

Here we again used rules of a functional derivative described in Appendix.