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Longer titles found: Upper-convected time derivative (view)

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alternate case: Time derivative

Material derivative (1,919 words) [view diff] exact match in snippet view article find links to article

{\mathbf {x} }}=\mathbf {0} } is chosen, the time derivative becomes equal to the partial time derivative, which agrees with the definition of a partial

i {\displaystyle x_{i}} which themselves depend on time. Then, the time derivative of L {\displaystyle L} is d L d t = d d t L ( t , x 1 ( t ) , … , x

map α∗ : TJ → TM so that the following diagram commutes: Then the time derivative α′ is the composition α′ = α∗ o ι, and α′(t) is its value at some point t ∈ J

average velocity approaches the instantaneous velocity, defined as the time derivative of the position vector, v = lim Δ t → 0 Δ r Δ t = d r d t = v x x ^

second time derivative is of interest in physics, as it is found in equations of motion for classical mechanical systems. The second time derivative of a

is the rate with respect to time at which work is done; it is the time derivative of work: P = d W d t , {\displaystyle P={\frac {dW}{dt}},} where P

vs. time graph), so the displacement is the rate of change (first time-derivative) of the absement. The dimension of absement is length multiplied by

\rho \,\mathrm {d} V}.} The conservation of mass requires that the time derivative of the mass inside a control volume be equal to the mass flux, J, across

the neutron star. PSR J0901–4046's period, combined with its period time derivative of 2.25×10−13 second/second, implies a characteristic age of 5.3 million

strictly positive for y ≠ 0 {\displaystyle y\neq 0} , and for which the time derivative V ˙ = ∇ V ⋅ g {\displaystyle {\dot {V}}=\nabla {V}\cdot g} is non positive

frame of reference. Acceleration is the second time derivative of position, or the first time derivative of velocity a i   = d e f   ( d 2 r d t 2 ) i

the Newmark- β {\displaystyle \beta } method states that the first time derivative (velocity in the equation of motion) can be solved as, u ˙ n + 1 =

Quantum thermodynamics is the study of the relations between two independent physical theories: thermodynamics and quantum mechanics. The two independent

solutions were nonphysical, and thus postulated that not only was the time derivative of the above expression equal to zero, but also that the expression

Acceleration a→ Rate of change of velocity per unit time: the second time derivative of position m/s2 L T−2 vector Action S Momentum of particle multiplied

variation. In most models, the secular variation is the amortized time derivative of the magnetic field B {\displaystyle \mathbf {B} } , B ˙ {\displaystyle

viewed by an observer moving with the ensemble (i.e., the convective time derivative is zero). Thus, if the microstates are uniformly distributed in phase

is the stress, η is the viscosity of the material, and dε/dt is the time derivative of strain. The relationship between stress and strain can be simplified

that involve the time-dependent deformation of a body often require a time derivative of the deformation gradient to be calculated. A geometrically consistent

{x} }{dt}}=g(t),} where v is velocity, x is position, d/dt is the time derivative, and g(t) may for instance be white noise. Since velocity changes instantly

named after Austrian theoretical physicist Paul Ehrenfest, relates the time derivative of the expectation values of the position and momentum operators x

physics, acceleration (symbol: a) is defined as the rate of change (or time derivative) of velocity. It is thus a vector quantity with dimension length/time2

calculus to deforming manifolds. Central to the CMS is the Tensorial Time Derivative ∇˙{\displaystyle {\dot {\nabla }}} whose original definition was put

relativistic version (with D'Alembertian instead of Laplacian and first-order time derivative) in 1969.. It is a classical wave equation with applications to extensions

}{\partial t}}\,,} in which the overdot denotes a partial time derivative ∂/∂t, not a total time derivative d/dt. For many fields φi(x, t) and their conjugates

reference configuration: it is constant in material coordinates. The time derivative of an integral over a volume is defined as d d t ( ∫ Ω ( t ) f ( x

{\displaystyle r} . r ¨ {\displaystyle {\ddot {\mathbf {r} }}} is the second time derivative of r {\displaystyle \mathbf {r} } . (the acceleration) G {\displaystyle

Here ∂ / ∂ t {\displaystyle \partial /{\partial t}} is the partial time derivative and ∇ 2 {\displaystyle \nabla ^{2}} is the Laplacian in n {\displaystyle

_{0}{\frac {\partial \mathbf {E} }{\partial t}}\times \mathbf {B} } The time derivative can be rewritten to something that can be interpreted physically, namely

\rangle } is an ensemble average and the dot above the A denotes the time derivative. For times t {\displaystyle t} that are greater than the correlation

of the integral of the equilibrium time correlation function of the time derivative of a corresponding microscopic variable A {\displaystyle A} (sometimes

{q_{r}}}} is an arbitrary generalized acceleration, or the second time derivative of the generalized coordinates q r {\displaystyle q_{r}} , and Q r

i.e. velocity, and x ¨ {\displaystyle {\ddot {x}}} is the second time-derivative of x , {\displaystyle x,} i.e. acceleration. The numbers δ , {\displaystyle

function is Ψ=ei(kx−ωt){\displaystyle \Psi =e^{i(kx-\omega t)}} The time derivative of Ψ is ∂Ψ∂t=−iωei(kx−ωt)=−iωΨ.{\displaystyle {\frac {\partial \Psi

{\displaystyle E} . These functions are called stationary states, because the time derivative at any point x {\displaystyle x} is always the same, so the amplitude

change over time. Newton's second law, in modern form, states that the time derivative of the momentum is the force: F → = d p → d t . {\displaystyle {\vec

}{T}}\right)+{\frac {\rho ~s}{T}}} where η ˙ {\displaystyle {\dot {\eta }}} is the time derivative of η {\displaystyle \eta } and ∇ ⋅ ( a ) {\displaystyle {\boldsymbol

quantity A {\displaystyle A} is a constant of the motion if its total time derivative is zero 0 = d A d t = ∂ A ∂ t + { A , H } , {\displaystyle 0={\frac

particle. Assuming that the masses are constant, G is one-half the time derivative of this moment of inertia 1 2 d I d t = 1 2 d d t ∑ k = 1 N m k r k

of the junction. To solve the above equation, first calculate the time derivative of the order parameter in superconductor A: ∂ ∂ t ( n A e i ϕ A ) =

such as time t, then the equation above reduces to the first ordinary time derivative of a with respect to t, d a d t = ∑ i = 1 n d a i d t e i . {\displaystyle

derivatives of the momenta of the system." By Newton's second law, the first time derivative of momentum is the force. The momentum of the i {\displaystyle i} -th

relationship between Entropy and Complexity as the latter being the time derivative of the former. He currently lives in Lugano, Switzerland. He went to

Banach space-valued functions, which provides a criterion for moving a time derivative of a function out of its action (as a functional) on the function itself

constant, and a ¨ {\displaystyle {\ddot {a}}} is the second proper time derivative of the scale factor. If we define (what might be called "effective")

density (current), ρ ˙ {\displaystyle {\dot {\rho }}} is the material time derivative of ρ {\displaystyle \rho } , v ( x , t ) {\displaystyle \mathbf {v}

the distribution of heat energy u in Rn. Indicating by ut (x, t) the time derivative of u(x, t), the initial value problem is { u t ( x , t ) − Δ u ( x

time dependent. It gets reflected in the governing equations as the time derivative of the properties are absent. For Studying Finite-volume method for

passband. A nth-order high-pass filter approximately applies the nth time derivative of signals whose frequency band is significantly lower than the filter's

{\displaystyle q_{0}} from the other measured parameters as above). The time derivative of the Hubble parameter can be written in terms of the deceleration

the acceleration of a particle moving along a curve in space is the time derivative of its velocity. In most applications, the acceleration vector is expressed

exists a ‖ u ‖ < ε {\displaystyle \|u\|<\varepsilon } so that the time derivative of the system's Lyapunov function is negative definite at that point

particle, the integral in the first term of the preceding equation, sans time derivative, is the probability of obtaining a value within V when the position

law from Maxwell's equations and the Lorentz force law. Consider the time-derivative of flux through a possibly moving loop, with area Σ ( t ) {\displaystyle

mass times it's acceleration, which is the time derivative of velocity and thus the second time derivative of spatial position. Starting from knowing

example, the 4-acceleration A μ {\displaystyle A^{\mu }} is the proper-time derivative of the 4-velocity U μ {\displaystyle U^{\mu }} : d d τ U = ( U ⋅ ∂

add a correction factor, which is generally done by subtracting the time derivative of the A vector potential described below. Whenever the charges are

timescale much slower than the scale of interest. This allows the time derivative in the momentum balance equation to be neglected. The ion temperature

Bosanquet equation is a differential equation that is second-order in the time derivative, similar to Newton's Second Law, and therefore takes into account the

relative to frame A, and so cannot include rotation of frame B. Taking a time derivative, the velocity of the particle is: d x A d t = d X A B d t + ∑ j = 1

electromagnetic induction. This law, in its modern form, states that the full-time derivative of the magnetic flux through a closed circuit induces an electromotive

deformed) state in the step. Here the superior dot represents the material time derivative ( ∂ / ∂ t {\displaystyle \partial /\partial t} following a given material

{L} =I{\boldsymbol {\omega }}} which means that the torque (i.e. the time derivative of the angular momentum) is τ = d I d t ω + I d ω d t . {\displaystyle

distance) travelled until time t {\displaystyle t} , the speed equals the time derivative of s {\displaystyle s} : v = d s d t . {\displaystyle v={\frac {ds}{dt}}

linear momentum and r is the position vector from the origin. The time-derivative of this is: d L d t = r × d p d t + d r d t × p . {\displaystyle {\frac

calculated from the particle interaction potentials; the dot is a time derivative such that X ˙ {\displaystyle {\dot {\mathbf {X} }}} is the velocity

[M]} is the mass matrix, [U¨]{\displaystyle [{\ddot {U}}]} is the 2nd time derivative of the displacement [U]{\displaystyle [U]} (i.e., the acceleration)

is positive for all r ≠ 0 {\displaystyle r\neq 0} . Now taking the time derivative of V {\displaystyle V} V ˙ = r r ˙ {\displaystyle {\dot {V}}=r{\dot

fluid V through a surface per unit time t. Since this is only the time derivative of volume, a scalar quantity, the volumetric flow rate is also a scalar

function [td, j] = lorenz_dynamics(state) % return a time derivative and a Jacobian of that time derivative x = state(1); y = state(2); z = state(3); sigma

{\displaystyle r'(t)=r(t)-vt.\,} The velocity of the particle is given by the time derivative of the position: u ′ ( t ) = d d t r ′ ( t ) = d d t r ( t ) − v =

fluid density, t is time, u is the flow velocity vector field. The time derivative can be understood as the accumulation (or loss) of mass in the system

V}{\partial q_{i}}}=0,~~~~i=1,2,...,n.} Here, the dot notation for the time derivative is used, d q d t = q ˙ {\displaystyle {\frac {dq}{dt}}={\dot {q}}}

then H, considered as a matrix function of X and P, will have zero time derivative. dHdt=P∗dPdt+(X+3εX2)∗dXdt=0 ,{\displaystyle {dH \over dt}=P*{dP \over

(\rho {\vec {v}})=0} The condition of incompressibility means that the time derivative of the density is 0, and that the density can be pulled out of the

types of terms: those involving the time derivative and those involving spatial derivatives. The time derivative portion is denoted as the local derivative

{\displaystyle u\in L^{2}\left([0,T];H_{0}^{1}(\Omega )\right)} with time derivative ∂ u ∂ t ∈ L 2 ( [ 0 , T ] ; H − 1 ( Ω ) ) . {\displaystyle {\frac {\partial

equation by using the method of false transients. In this method, a time derivative of the dependent variable is added to Laplace’s equation. Finite differences

Coulomb's law for the static electric field. The second term is the time derivative of the first Coulombic term multiplied by r ′ c {\displaystyle {\frac

the diagram below from the previous Phase Difference tutorial. The time derivative or integral of a phasor produces another phasor. For example: Re ⁡

Velocity is the time derivative of position. Its dimensions are length/time. Acceleration a of a point is vector which is the time derivative of velocity

means that it will have a non-zero first, and also a non-zero second time derivative (this is of course true regardless the choice of the coordinate system)

{\displaystyle \Delta t} tend to zero, that is, the velocity is the time derivative of the displacement as a function of time. v = lim Δ t → 0 Δ x Δ t

See final chapter and appendix. Sutton, R. S.; Barto, A. G. (1990). "Time Derivative Models of Pavlovian Reinforcement" (PDF). Learning and Computational

t}}.} This has again the form of the Schrödinger equation, with the time derivative of the wave function being given by a Hamiltonian operator acting upon

function of q alone (i.e., T and V are scleronomic). In this example, the time derivative of q is the velocity, and so the first Hamilton equation means that

burn the mass of the spacecraft will decrease due to use of fuel, the time derivative of the mass being If now the direction of the force, i.e. the direction

disturbances in space-time, caused by the motion of masses, if the third time derivative of the mass quadrupole moment is non-zero. These waves are very weak

x , t ) {\displaystyle y(x,t)} , and the coefficient of the second time derivative term is equal to 1 v 2 {\displaystyle {\frac {1}{v^{2}}}} ; thus v

1088/0264-9381/21/11/006. ISSN 0264-9381. S2CID 250859930. Snap [the fourth time derivative] is also sometimes called jounce. The fifth and sixth time derivatives

energy is defined in terms of the energy operator (Hamiltonian) as a time derivative of the wave function. The Schrödinger equation equates the energy operator

y,z,t)=\psi (x,y,z)e^{i\omega t}.} Substituting and evaluating the time derivative gives ( ∇ 2 + ω 2 c 2 ) ψ ( x , y , z ) = 0. {\displaystyle \left(\nabla

physical significance of this function is understood by taking its total time derivative d W d t = ∂ W ∂ q i q ˙ i = p i q ˙ i . {\displaystyle {\frac {dW}{dt}}={\frac

a{\displaystyle a}. One can see this invariance by considering the time derivative ddt(y−x21+2a){\displaystyle {\tfrac {\mathrm {d} }{\mathrm {d} t}}\left(y-{\tfrac

order approximated by a strain tensor that changes with time. The time derivative of that tensor is the strain rate tensor, that expresses how the element's

kinematics equations, which yields formulas for the joint parameters. The time derivative of the kinematics equations yields the Jacobian of the robot, which

to 0. This similarly applies for the density perturbation times the time derivative of the velocity. Moreover, the spatial components of the material derivative

simulation, by a corresponding steady-state simulation (where the time derivative in the groundwater flow equation is set equal to 0). There are two

}}}+z\,{\boldsymbol {\hat {z}}}.} The velocity of the particle is the time derivative of its position, v = d r d t = ρ ˙ ρ ^ + ρ φ ˙ φ ^ + z ˙ z ^ , {\displaystyle

{\displaystyle \Phi _{B}} (and therefore the electromotive force) by taking the time derivative of the flux relation E = − d Φ B d t = B A sin ⁡ θ d θ d t {\displaystyle

resistor must be equal in magnitude (but opposite in sign) to the time derivative of the accumulated charge on the capacitor. This results in the linear

Birkhoff's theorem. More technically, the second time derivative of the quadrupole moment (or the l-th time derivative of the l-th multipole moment) of an isolated

of the rotating frame (i.e. P = P1 i + P2 j +P3 k), then the first time derivative [dP/dt] of P with respect to the rotating frame is, by definition,

or the dot notation, a dot is placed over a symbol to represent a time derivative. If y {\displaystyle y} is a function of t {\displaystyle t} , then

{\displaystyle \mathbf {a} } is the acceleration vector, the second time derivative of the position vector r {\displaystyle \mathbf {r} } Strictly speaking

H(t)\equiv {{\dot {a}}(t) \over a(t)}} where the dot represents a time derivative. The Hubble parameter varies with time, not with space, with the Hubble

x')-\mathbb {P} _{t,t'}(x\mid x')\right)\,dx,\end{aligned}}} which is a time derivative. Finally we arrive to ∫ [ L p ( x ) ] P t , t ′ ( x ∣ x ′ ) d x = ∫

the velocity gradient along the streamline. For a steady flow, the time derivative of the velocity is zero: ∂ c ∂ t = 0 {\displaystyle {\frac {\partial

{\displaystyle R} , the function r ( τ , R ) {\displaystyle r(\tau ,R)} and its time derivative respectively provides its law of motion and radial velocity. An interesting

in first-order spatial derivatives of the wave solution, while the time-derivative on the other side of the equations, which gives the other field, is

in first-order spatial derivatives of the wave solution, while the time-derivative on the other side of the equations, which gives the other field, is

approximately by the equivalent of a low-pass EMA filter of 38 days. The time derivative estimate (per day) is the MACD value divided by 14. The average series

_{0}^{2\pi }p{\frac {\partial x}{\partial \theta }}\,d\theta .} The time derivative of this quantity is d J d t = ∫ 0 2 π ( d p d t ∂ x ∂ θ + p d d t ∂

where: X ˙ {\displaystyle {\dot {X}}} is the velocity, the dot being a time derivative U ( X ) {\displaystyle U(X)} is the particle interaction potential

It is surprising that the Hamiltonian density doesn't depend on the time derivative of ψ {\displaystyle \psi } , directly, but the expression is correct

matrix is not time-dependent, and the relative sign ensures that the time derivative of the expected value ⟨ A ⟩ {\displaystyle \langle A\rangle } comes

rate of the Cauchy stress is the Piola transformation of the material time derivative of the 2nd P-K stress. We thus define σ ∘ = J − 1   ϕ ∗ [ S ˙ ] {\displaystyle

is simply zero. The four-acceleration is proportional to the proper time derivative of the four-momentum divided by the particle's mass, so p μ A μ = η

\ldots ,q_{N-1},q_{N})} A dot over a variable or list signifies the time derivative (see Newton's notation). For example, q ˙ = d q d t . {\displaystyle

the mass density, ρ˙{\displaystyle {\dot {\rho }}} is the material time derivative of ρ{\displaystyle \rho }, v(x,t)=u˙(x,t){\displaystyle \mathbf {v}

instead – like a partial differential equation involving a second-order time derivative. Some models of nonlinear heat conduction (which are also parabolic

given coordinate qi does not appear in the Lagrangian (although its time derivative might appear), then pj is constant. This is the generalization of the

p_{N}\right).\end{aligned}}} A dot over a variable or list signifies the time derivative, e.g., q ˙ ≡ d q d t {\displaystyle {\dot {\mathbf {q} }}\equiv {\frac

both gas and porous medium properties). For very short time scales, a time derivative of flux may be added to Darcy's law, which results in valid solutions

where R x → y {\displaystyle \scriptstyle R_{x\rightarrow y}} is the time derivative of the K matrix: R x → y = K x → y d t − δ x y d t . {\displaystyle

Q^{\dagger }\,\right\}=2i{\frac {\partial }{\partial t}}} where i times the time derivative is the Hamiltonian operator in quantum mechanics. Both Q and its adjoint

y=x−x3/3−x˙/μ{\displaystyle y=x-x^{3}/3-{\dot {x}}/\mu }, where the dot indicates the time derivative, the Van der Pol oscillator can be written in its two-dimensional form:

described with a linear differential equation that is first-order in the time derivative, which is generally the case in order for something to be validly called

momentum P ^ {\displaystyle {\hat {P}}} are all linear operators. The time derivative of a quantum state is - i H ^ / ℏ {\displaystyle i{\hat {H}}/\hbar

velocity of the COM frame. Since V is the velocity of the COM, i.e. the time derivative of the COM location R (position of the center of mass of the system):

time, is denoted by Q ˙ {\displaystyle {\dot {Q}}} , but it is not a time derivative of a function of state (which can also be written with the dot notation)

electrical conductivity σ and the permeability μ . The term ∂B/∂t is the time derivative of the field; ∇2 is the Laplace operator and ∇× is the curl operator

asymmetric: infinite-order in space derivatives but only first order in the time derivative, which is inelegant and unwieldy. Again, there is the problem of the

inflation is the derivative of inflation, opposite in sign to the second time derivative of the purchasing power of money. Stating that a function is decreasing

displacements is obtained by integrating twice (acceleration is the second time derivative of displacement) the difference of the applied acceleration and the

{m}{L_{y}}}\right)^{2}}}.} If the initial conditions for z(x,y,0) and its time derivative ż(x,y,0) are chosen so the t-dependence is a cosine function, then

also functions of time. Let x˙{\displaystyle {\dot {x}}} denote the time derivative of the constant mode x{\displaystyle x}. This represents a type of

P_{n})^{\mathrm {T} }} are new canonical coordinates, then, with a dot denoting time derivative, Z ˙ = M ( z , t ) z ˙ , {\displaystyle {\dot {\mathbf {Z} }}=M({\mathbf

epoch. The definition of "accelerating expansion" is that the second time derivative of the cosmic scale factor, a ¨ {\displaystyle {\ddot {a}}} , is positive

terms as well, but the extra terms in the Lagrangian add up to a total time derivative of a scalar function, and therefore still produce the same Euler–Lagrange

region recovered. Rate of change of frequency (also RoCoF) is simply a time derivative of the utility frequency ( d f / d t {\displaystyle {df}/{dt}} ), usually

+2\left\langle \varepsilon (t)\right\rangle ^{2}\right)\end{aligned}}} Now the time derivative of the second moment: d ⟨ ε 2 ⟩ d t = l i m d t → 0 ⟨ ε 2 ( t + d t

electric fields, as is the case with conventional probes sensitive to the time-derivative of the electric field. Electro-optic measurements of strong electromagnetic

Here, q ˙ ( t ) {\displaystyle {\dot {\boldsymbol {q}}}(t)} is the time derivative of q ( t ) . {\displaystyle {\boldsymbol {q}}(t).} When we say stationary

dependent on time and the product rule is needed to compute the total time derivative d d t m r 2 φ ˙ {\textstyle {\frac {d}{dt}}mr^{2}{\dot {\varphi }}}

an approach that respects the meanings of all terms is to take the time derivative of the equation to eliminate ⟨R⟩0 and t. Another such approach is to

{\vec {B}}\right)} and using the general expression for the co-moving time derivative d v → s d t = ∂ v → s ∂ t + 1 2 ∇ → v s 2 − v → s × ( ∇ → × v → s )

Wojnar. In 1994, Elishakoff suggested to neglect the fourth-order time derivative in Uflyand-Mindlin equations. A similar, but not identical, theory

position vector of the charged particle, t is time, and the overdot is a time derivative. A positively charged particle will be accelerated in the same linear

e^{-Bt}+C_{2}\cdot e^{-At}+{\frac {Q_{t}\cdot e^{-At}-R_{t}\cdot e^{-Bt}}{P}}} with time derivative: dxdt=−Ae−At⋅C2−Be−Bt⋅C1+1P[Qt˙⋅e−At−AQt⋅e−At−Rt˙⋅e−Bt+BRt⋅e−Bt]{\displaystyle

}(t)|^{2}} divided by d t , {\displaystyle \mathrm {d} t,} which is the time derivative of | c ν ( t ) | 2 , {\displaystyle |c_{\nu }(t)|^{2},} Γ μ → ν   =

term in the Landau–Lifshitz (LL) equation by one that depends on the time derivative of the magnetization: This is the Landau–Lifshitz–Gilbert (LLG) equation

difference for the time derivative and central difference for the spatial derivatives. For both momentum equations, the time derivative becomes ∂ u i ∂ t

= LP - PL is the Lie commutator of the two operators) is equivalent to the time derivative of Flaschka's variables. The choice L ( t ) f ( n ) = a ( n , t ) f

the Lagrangian density, ϕ ˙ {\displaystyle {\dot {\phi }}} is the time-derivative of the field, ∇ is the gradient operator, and m is a real parameter

of matter entering the system. It should not be confused with the time derivative of the entropy. If matter is supplied at several places we have to

{\displaystyle {\frac {dK(t)}{dt}}} , the derivative with respect to time. Derivative with respect to time means that it is the change in capital stock—output

{\displaystyle \mathbf {L} } . Specifying to inverse-square central forces, the time derivative of p × L {\displaystyle \mathbf {p} \times \mathbf {L} } is d d t p

For the case where the source is an unchanging voltage, taking the time derivative and dividing by L leads to the following second order differential

where m is the invariant mass. The four-acceleration is the proper time derivative of 4-velocity: A μ = d U μ d τ . {\displaystyle A^{\mu }={\frac {dU^{\mu

change in polarization during a continuous physical process is the time derivative of the polarization and also can be formulated in terms of the Berry

integration in time removes the time dependence of the resultant terms, the time derivative must be eliminated, leaving: ρ u ¯ j ∂ u ¯ i ∂ x j = ρ f i ¯ + ∂ ∂

easy to recall where factors of c go in the Maxwell equation. Every time derivative comes with a 1 / c, which makes it dimensionally the same as a space

(mc^{2})^{2}\phi .} Applying this yields the non-relativistic limit of the second time derivative of ψ {\displaystyle \psi } , ∂ ψ ∂ t = ( − i m c 2 ℏ ϕ + ∂ ϕ ∂ t )

right plot shows the time derivative of the restricted solution as a function of time (blue curve), as well as the time derivative dU/dt{\displaystyle

corresponding s m ( t ) {\displaystyle s_{\mathrm {m} }(t)} . The time derivative of the unwrapped instantaneous phase has units of radians/second, and

The above inner product can also be written in terms of β and its time derivative. Then the relativistic generalization of the Larmor formula is (in

structure of this target system, we use the Lie derivative. Consider the time derivative of (2), which can be computed using the chain rule, y ˙ = d h ( x )

permittivity, β ˙ {\displaystyle {\dot {\boldsymbol {\beta }}}} signifies a time derivative of β {\displaystyle {\boldsymbol {\beta }}} , and q is the charge of

set to unity. In this unit system, they are: Above, a dot denotes a time derivative and the axion–photon coupling is g a γ γ {\displaystyle g_{a\gamma

representation theorem can be interpreted formally as saying that α is the "time derivative" of M with respect to Brownian motion B, since α is precisely the process

A ) i j {\displaystyle (V_{R})_{j}^{i}-(V_{A})_{i}^{j}} is a total time derivative, i.e. a divergence in the calculus of variations, and thus it gives

centered on the field. These induced currents are proportional to the time derivative of the magnetic field, leading it by 90 degrees. This puts the eddy

{\frac {\dot {a}}{a}}\!} where a ˙ {\displaystyle {\dot {a}}} is the time-derivative of the scale factor. The Friedmann equations express the expansion

the state change over one moment of time, then we can state that the time derivative of n i {\displaystyle n_{i}} is given by this difference, n ˙ i = n

while z{\displaystyle z} is known as the algebraic variable. The time derivative of the differential variable, y˙{\displaystyle {\dot {y}}}, depends

|\mathbf {L} |=mr^{2}\omega =m\ell ^{2}{\frac {d\theta }{dt}}.} and its time derivative d d t | L | = m ℓ 2 d 2 θ d t 2 , {\displaystyle {\frac {d}{dt}}|\mathbf

{x^{2}}{2A^{2}}}}\right)\cdot J,} a state with approximately vanishing time derivative is produced, ‖ d d t Q A | 0 ⟩ ‖ ≈ 1 A ‖ Q A | 0 ⟩ ‖ . {\displaystyle

where the double dot on u ¨ {\displaystyle {\ddot {u}}} denotes double time derivative of u, ∇ is the nabla operator, and ∇2 = ∇ · ∇ is the (spatial) Laplacian

{dV_{\text{in}}}{dt}}.} Consequently, The output voltage is proportional to the time derivative of the input voltage with a gain of R C . {\displaystyle RC.} Hence

{a} _{i}\cdot {\dot {\mathbf {r} }}_{i},} where the dot represents a time derivative and a i {\displaystyle \mathbf {a} _{i}} are vector coefficients depending

}^{s}} respectively. Taking the time-derivative of this function allows for some simplifications. The time-derivative of one of the terms in eq. (7) is

systems, some characteristic of control quality is used (e.g., the first time derivative of a controlled parameter). Automatic tuning makes sure that this characteristic

γ=1/√(1-v^2/c^2), the Lorentz factor. This is easily shown by taking the second time derivative of that dot product. Because this angular frequency exceeds ω, the

characterized by using fractional-order diffusion equation models. The time derivative term corresponds to long-time heavy tail decay and the spatial derivative

threshold is not a linear acceleration but rather a jerk motion (third time derivative of position), and the reported threshold value is on the order of 0

Then, given conservation of mass, the continuity equation relates the time derivative of the concentration with the divergence of the flux: ∂ ρ ( x , t )

or its derivatives. The self-adjoint operator L{\textstyle L} has a time derivative Lt{\textstyle L_{t}} and generates a eigenvalue (spectral) equation

equal to the centripetal acceleration. The moment equation is the time derivative of the angular momentum: M = B d 2 θ d t 2 {\displaystyle M=B{\frac

operator T ^ {\displaystyle {\hat {\mathbf {T} }}} is obtained from the time derivative of the spin operator: T ^ = d S ^ d t = − i ℏ [ ℏ 2 σ , H ^ ] {\displaystyle

time-invariant and the 'der' operator, which represents (symbolically) the time derivative of a variable. Also worth noting are the documentation strings that

quasi-momentum: the added mass times the body acceleration is equal to the time derivative of the fluid quasi-momentum. Unsteady forces due to a change of the

{\frac {\dot {a}}{a}},} where a ˙ {\displaystyle {\dot {a}}} is the time-derivative of the scale factor. The first Friedmann equation gives the expansion

is either 1 if the system at time t is in state X or 0 if not. The time-derivative C'(t) starts at time 0 at the transition state theory (TST) value kABTST

f_{n}(t)\ } are very nearly harmonic but not necessarily exactly so. The time-derivative of φ n ( t )   {\displaystyle \varphi _{n}(t)\ } , that is φ n ′ (

{d}{dt}}\,{\vec {v}}=-{\vec {\nabla }}(V+Q)~,} where the hydrodynamic time derivative is defined as d d t = ∂ ∂ t + v → ⋅ ∇ →   . {\displaystyle {\frac {d}{dt}}={\frac

{\partial u}{\partial t}}+{\frac {\partial p}{\partial x}}=0} . Taking the time derivative of the continuity equation and the spatial derivative of the force

Timoshenko and P. Ehrenfest overestimate the importance of the fourth-order time derivative in their theory of beams?", Journal of Vibration and Acoustics, Vol

solve. A final assumption is the Born-Markov approximation that the time derivative of the density matrix depends only on its current state, and not on

space variables only and ω ˙ {\displaystyle {\dot {\omega }}} is the time derivative of ω. If we move into spacetime M = R × R 3 {\displaystyle M=\mathbb

and an electric inductance switched in series. It was shown that the time derivative of the heat current is proportional to the negative temperature difference

\eta \gamma ^{2}\omega /\pi (\gamma ^{2}+\omega ^{2})} and taking the time derivative of the path integral form density matrix the equation and writing it

voltage traces, as is the case for conductive probes sensitive to the time-derivative of the electric field. EOMs can be based on many operating principles

_{r}=4\pi rf} where ∇ t {\displaystyle \nabla _{t}} is the covariant time derivative (and ∇ {\displaystyle \nabla } the Levi-Civita connection), p {\displaystyle

principle of extremal action, see electromagnetic tensor. Often, the time derivative in the Faraday–Maxwell equation motivates calling this equation "dynamical"

charge (x = q) and since the charge is related to the current via the time derivative dq/dt = i(t). Thus for pure memristors f (i.e. the rate of change of

either diminishes or leaves unchanged the norm of its operand). If the time derivative of a point x in the space is given by Ax, then the time evolution is

\right)\right].\\\end{aligned}}} The second term in the last equality is the time derivative of a quantity that is related through a multiplicative constant to

that the velocity and pressure do not vary with time), we can set the time derivative terms equal to zero: ( v ⋅ ∇ ) v = − 1 ρ ∇ p + ν Δ v {\displaystyle

the layers. Stress tensor (disambiguation) Finite strain theory § Time-derivative of the deformation gradient, the spatial and material velocity gradient

t}+(1-f)\cdot F_{\varphi }^{t}]\,\Delta t} After discretizing the time derivative, function F ( φ ) {\displaystyle F(\varphi )} remains to be evaluated

Defining Δ = ω f i − ω {\displaystyle \Delta =\omega _{fi}-\omega } the time derivative of sin 2 ⁡ ( Δ t / 2 ) / Δ 2 {\displaystyle \sin ^{2}(\Delta t/2)/\Delta

examined, it can be seen that the change in the E-field in time (the time derivative) is dependent on the change in the H-field across space (the curl)

objects moving through four-dimensional space appears from the regular time derivative c 2 = ( d x / d t ) 2 + ( d y / d t ) 2 + ( d z / d t ) 2 + ( c d τ

{\frac {DA_{e}}{Dt}},} (1) where DDt{\textstyle {\frac {D}{Dt}}} is the time derivative in the rotational frame (not inertial frame), C{\displaystyle C} is

as h˙=2πTs{\displaystyle {\dot {h}}={\frac {2\pi }{T_{s}}}} and the time derivative of the tan q expression is q˙1cos2⁡q=cos⁡φ[cos⁡hcos⁡δsin⁡φ−sin⁡δco

potential energy term. The kinetic energy term involves the square of the time derivative of y ( x , t ) {\displaystyle y(x,t)} and thus gains a factor of ω

inner product. Rewriting the limits in terms of the integral of a time derivative, we have M=−1Z∫d4x1∂0(eip1⋅x1⟨β out|Ψ¯(x1)γ0up1s1|α′ in⟩){\displaystyle

be shown that ω{\displaystyle {\boldsymbol {\omega }}} is not the time derivative of any vector, in contrast to the usual definition of velocity. The

generic Langevin equation is obtained by applying this equation to the time derivative of a slow variable A{\displaystyle A}, dA(Γ,t)/dt=eitL(dA(Γ,t)/dt)t=0{\displaystyle

relation is given by the convolution of the capacitance with the first time-derivative of the voltage, i.e., Q(t)=C(t)∗V˙(t){\displaystyle Q(t)=C(t)*{\dot

then integrating twice to invert the effect of the double partial-time derivative. The analogue electronic circuit equivalents of a square root function

{\dot {X}}=\nabla \log \pi (X)+{\sqrt {2}}{\dot {W}}} driven by the time derivative of a standard Brownian motion W {\displaystyle W} . (Note that another

}{\frac {dV(t)}{dt}}=-\sum _{i}I_{i}(t,V).} The above equation is the time derivative of the law of capacitance, Q = CV where the change of the total charge

possible to get an expression for η {\displaystyle \eta } , by taking the time derivative of the continuity equation and substituting the momentum equation:

mechanics are all balance equations, and as such each of them contains a time-derivative term which calculates how much the dependent variable change with time

electron experiences must be zero given the connection between the time derivative of spatially averaged momentum, which vanishes as a bound state, and

per second. Following Lenz's law, the voltage is proportional to the time derivative of magnetic flux. The voltage will be amplified by the apparent permeability

Lagrangian is evaluated with q˙{\displaystyle {\dot {q}}} representing the time derivative of the state's evolution and p{\displaystyle p}, the so-called "conjugate

F=p˙{\displaystyle F={\dot {p}}}, where p˙{\displaystyle {\dot {p}}} is the time derivative of the momentum p{\displaystyle p}. This seems a trivial enough fact

{\displaystyle {\vec {v}}_{s}} is the velocity of the superfluid component. The time derivative is the so-called hydrodynamic derivative, i.e. the rate of increase

{\partial \mathbf {u} }{\partial t}}} In case of a steady flow the time derivative of the flow velocity disappears, so the momentum equation becomes:

time-dependent adiabatic Hamiltonian, and the overdot represents a time derivative. Comparison of the initial conditions used with the values of the state

{F}}^{e})^{-1}\,.\end{aligned}}} where a superposed dot indicates a time derivative. We can write the above as l = l e + F e ⋅ L p ⋅ ( F e ) − 1 . {\displaystyle

shape similar to an arterial pressure waveform. The calculated first time derivative of dZ(t) is the d Z ( t ) d t {\displaystyle {\frac {dZ(t)}{dt}}} waveform

between two kernels into strain (deformation). Strain rate will then be time derivative of strain. In some commercial applications, the acoustic markers are

{\partial \mathbf {u} }{\partial t}}} In case of a steady flow the time derivative of the flow velocity disappears, so the momentum equation becomes:

seems naively to be overcounting. This is not the case, because the time derivative term in L includes the overlap between the different field states.

A(t), using a Poincaré−Hausdorff matrix identity, he could relate the time derivative of Ω to the generating function of Bernoulli numbers and the adjoint

chirality was offered in 2012; optical chirality is to the curl or time derivative of the electromagnetic field what helicity, spin and related quantities

{\displaystyle i} . The acceleration matrix can be defined as the sum of the time derivative of the velocity plus the velocity squared H i , j ( k ) = W ˙ i , j

In general, the angular velocity in an n-dimensional space is the time derivative of the angular displacement tensor, which is a second rank skew-symmetric

^{2}=-{\frac {\hbar }{2m}}} and H j j = 0. {\displaystyle H_{jj}=0.} The time derivative term in the transition equation can be identified with the energy of

}}^{2}+ml(g+a~\nu ^{2}\cos \nu t)\cos \varphi ,} up to irrelevant total time derivative terms. The differential equation φ ¨ = − ( g + a   ν 2 cos ⁡ ν t )

{\displaystyle u} in the zonal direction, the time derivative of the mean flow U {\displaystyle U} and the time derivative of β y {\displaystyle \beta y} are zero

fingering instability in gravity driven flow. The lack of a second-order time derivative in the thin-film equation is a result of the assumption of small Reynold's

the integrated densitized trace of the extrinsic curvature is the "time derivative of the volume". Misner, Charles W.; Thorne, Kip S.; Wheeler, John Archibald

While spatial derivatives are equal in both coordinate systems, the time derivative that appears in the equations satisfies: ∂ u → ( x → , t ) ∂ t = ∑

either the radial or axial momentum equations (after removing the time derivative term) to solve for ω θ {\displaystyle \omega _{\theta }} . For instance

equation of motion governing the rotation of the body is derived from the time derivative of angular momentum: N=Cd2ψdt2{\displaystyle N=C{\frac {d^{2}\psi }{dt^{2}}}}

consisting of Navier stokes equation and continuity equations with time derivative of pressure, then the solution will be same as the stationary solution

S-equation. A further generalization is obtained by replacing the first time derivative by a Riesz fractional derivative yielding a time-fractional S-equation

(February 2009). "Optimization of mixed quantum-classical dynamics: Time-derivative coupling terms and selected couplings". Chemical Physics. 356 (1–3):

Differentiation was according to the product rule, while ∂A/∂t is the time derivative of the initial A, not the A(t) operator defined. The last equation

speed is termed "mechanical impedance". Note: Here, speed means the time derivative of a displacement, not the speed of sound. There also is an electro-acoustic

rotation matrix is time dependent, then it does not commute with the time derivative. In that case, the rotation of the separation velocity can be written

}}} . Except for the sign, the right-hand side corresponds to the time derivative of the kinetic energy. This means that every change of the potential

{\partial u}{\partial x}}=0.} Further, since (remember that the time derivative of a function f(x,t){\displaystyle f(x,t)} integrated along a curve

variable where the conjugate "momentum" is equal to capacitance times the time derivative of magnetic flux. The conjugate "momentum" is really the charge. π

doi:10.1016/0142-1123(84)90032-X. Pommier, S.; Risbet, M. (2005). "Time derivative equations for mode I fatigue crack growth in metals". International

{\displaystyle x} and y {\displaystyle y} directions, and a dot indicates a time derivative (i.e., x ˙ {\displaystyle {\dot {x}}} is the x {\displaystyle x} velocity

is significant, the equation is further simplified by removing the time derivative of the magnetic vector potential: ∇⋅[D(∇c+zekBTc∇ϕ)]=0{\displaystyle

quantity k in equations (1.1.1) and (1.1.2) comes originally from the time derivative in Maxwell's equations and is the free space propagation constant (actually

evidence to suggest that they respond to the force and yank (the first time-derivative of force) exerted on intrafusal muscle. Spindles are also composed

that the dot products Π (between π and the time derivative of z) and B (between E and the time derivative of z) must equal each other. They state: "Except

is the coordinates respective to the standard form. If we take the time derivative in both sides and invert the fundamental matrix we obtain x ˙ = ε e

{D}}h}(t,\omega )} is the short-time Fourier transform computed using a time-derivative analysis window h D ( t ) = d d t h ( t ) {\displaystyle h_{\mathcal

Where w ¨ {\displaystyle {\ddot {\mathbf {w} }}} denotes the double time derivative of w {\displaystyle \mathbf {w} } ϵ 0 {\displaystyle \epsilon _{0}}

}}c/V_{\infty }}}} , where α ˙ {\displaystyle {\dot {\alpha }}} is the time derivative of AoA, c {\displaystyle c} is the blade chord, and V ∞ {\displaystyle

defines the equations of motion and constraints of the system. The time derivative may be either prescribed or numerically approximated from the snapshots

been carried out by Baacke. The quantum energy has been derived as a time derivative of the one-loop fermion effective action S eff = − i ln ⁡ det ( i γ

names) of continuum mechanics, with another term being the partial time derivative: D D t = ∂ ∂ t + u ⋅ ∇ {\displaystyle {\frac {D}{Dt}}={\frac {\partial

often made in a moving frame of reference. To account for this, the time derivative of the observed variables in the fixed frame of reference (ρ, θ, z)

field resulting from the polarization is proportional to its second time derivative, i.e., E∝∂2∂t2P{\displaystyle {\mathbf {E} }\propto {\frac {\partial

n ) {\displaystyle z(t_{n})} . Thus, the BDF2 approximation of the time derivative reads as follows: ( ∂ u h ∂ t ) n + 1 ≃ 3 u h n + 1 − 4 u h n + u h

the integrated densitized trace of the extrinsic curvature is the``time derivative of the volume". The Lagrangian for a scalar field in curved spacetime

{v}})-\sigma _{ij}{\partial v_{i} \over {\partial x_{j}}}} Now expanding the time derivative of the total energy, we have: ∂ ∂ t [ ρ ( k + ε ) ] = ρ ∂ k ∂ t + ρ