WW is the volume of the fluid under consideration at t=0t=0 and WtWt is that volume at time tt. JJ is the Jacobian determinant of the transformation, followed by differenciating under the integral sign (RHS) ∂∂t(ρu)(ϕ(x,t),t)=(DDtρu)(ϕ(x,t),t)∂∂t(ρu)(...
Liénard’s generalisationIn this pedagogical article, we elucidate the direct derivation of total power emitted by an accelerating charged particle, known as Liénard's generalisation, using differentiation under integral sign technique.doi:10.1007/s12045-018-0662-7Biswas, Anindya Kumar...
The stochastic movement of the particle turns its classical action integral into a stochastic variable. 3. Nature tries to minimize the expected value for the action, in which the particle’s velocity is consider to be a control parameter of the optimization. The principles formulated above are...
An exception, note E geometrically derives the integral term in D'Alemberts formula which is used in the SI This derivation is originated here. With this I have shown the linkage between the analysis sections “Stationary planar source and elimination of the wake”, “Moving planar source and ...
This implies that, although the calcu- lations made in the DR-scheme do not produce the NSVZ equation, they nevertheless confirm that it is valid in a cer- tain (NSVZ) subtraction scheme. Using the general equa- tions describing how the NSVZ equation changes under finite renormalizations [...
The presence of exp(–2DλQ) under the integral of Eq. (18) significantly reduces the Bv contribution to the integral and thus to the permeability for the poor rock compared to good rock. Download: Download full-size image Fig. 4. The effect on the exponential of the Laplace ...
This leads to the expres- sion Z = 1 N [Dϕ][Dπ] exp − dt 1 2 (1 +i ϕ)π 2 −iπ ˙ ϕ −α(1 +i ϕ) 2 . (9) Replacing π˙ ϕ by - ˙πϕ under the t integration, we have an exponent that is quadratic in ϕ, so the ϕ functional ...
used supervised learning to derive an effective spin-1/2 Hamiltonian in the strong-coupling limit of a half-filled Hubbard model. In contrast to the scenario under our current consideration, this problem admits a well-behaved perturbative expansion, which can be used to corroborate the success of...
and calculate the integral over \(d^8y\) with the help of the \(\delta \) -function, $$\begin{aligned} \delta _{\mathrm{matter}}\bigg (\frac{\beta }{\alpha _0^2}\bigg )= & {} - \frac{1}{8\pi r{{\mathcal {v}}}_4} c(r)_i{}^j \left( \gamma _\phi \right) ...
The presence of exp(–2DλQ) under the integral of Eq. (18) significantly reduces the Bv contribution to the integral and thus to the permeability for the poor rock compared to good rock. Download: Download full-size image Fig. 4. The effect on the exponential of the Laplace ...