Sentence examples for be written as partial from inspiring English sources

The ϕ component of the Faraday equation can be written as partial _{t}B^{phi}-rB^{i}partial _{i} biggl(frac{v^{phi}}{r} biggr) + partial _{i} bigl v^{i} B^{phi}bigr) =0, (18) where (i={r,z}).

These can be written as partial _{t} {T^{tmu}} + partial _{z}T^{zmu} + nabla_{r}{T^{rmu}}=0, (24) so the steady-state versions are partial _{z}T^{zmu} + nabla_{r}{T^{rmu}}=0. (25) These already have the same form as the 1D time-dependent equations, so we only need to show that T^{zmu}simeq T^{tmu}.

Accordingly, the evolution of the dislocation density tensor is written as (partial _{t} {boldsymbol {alpha }} = sum _{varsigma }partial _{t} {boldsymbol {rho }}^{varsigma }otimes {boldsymbol {mathrm {b}}}^{varsigma }) with ∂ t ρ ς =∇×[ v ς ×ρ ς ] where the dislocation velocities v ς are again slip system specific.

The point defect balance and the phase field evolution equations can then be written as {partial}_t{c}_{alpha }={P}_{alpha }-nabla cdot {mathbf{J}}_{alpha }-{R}_{mathrm{iv}}, (17.1) {partial}_teta =- Lu. (17.2).

The ideal linear models (i.e., the theoretically true classification functions) can be written as: the partial model h p = Σ i = 1 p v i x i and the full model h p + q = Σ i = 1 p w i x i + Σ j = 1 q w p + j x p + j.

We demonstrate that the equations can be written as a coupled system of Partial Differential Equations (PDEs) comprising: (i) first order hyperbolic PDEs for the volume fraction of each phase; (ii) a generalised Stokes flow for the velocity of each phase; and (iii) elliptic PDEs for the concentration of nutrients and messengers.

The mixed partial can be written as follows P m ( t ) = A m ( t ) e j [ ϕ m ( t ) ] = ∑ s k P s k ( t ) = ∑ s k A s k ( t ) e j [ ϕ s k ( t ) ] (22).

The Lagrangian density (2.2) can be written as: begin{aligned} mathcal {L}&= partial ^{nu }phi ^(x partial _{nu }phi (x -m^{2}phi ^(x -m^{2}phinumber &quad -, gphi ^(x phi (x nonumber}(x) nonumber &quad +,{frac{1}{2}}int {rm{d}}{x^{prime }}rho (x)D(x-{x^{prime }})rho ({ x^{prime }}), end{aligphi} (2.4)where we have suppressed the free part of the Lagrangian in the above equation.

Once found α k, p k, and q k for each source k, the overlapping partial can be written as P m ( t ) = ∑ s k α k A wk ( t ) e j p k q k ϕ wk ( t ) (29). and the separated contributions of each present source are of course P s k ( t ) = α k A wk ( t ) e j p k q k ϕ wk ( t ) (30).

The first and the second invariant of effective stress partial tensor can be written as below.

Then, the partial derivatives can be written as follows: begin{array}{*{20}l} nabla_{boldsymbol{x x}}^{2} thinspace f &= 2h left(mathbf{1}tilde{boldsymbol{w}}^{intercal} + tilde{boldsymbol{w}} mathbf{1}^{intercal}right) + 2q tilde{boldsymbol{w}} tilde{boldsymbol{w}}^{intercal}, end{array} (14a).