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Substituting v < sub > 1 </ sub > into the identity and removing common factors gives the numerical example cited above.
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Substituting and v
Substituting from the expressions for v and v < sub > m </ sub > above gives:
Substituting and <
Substituting into the above formula for | F < sub > h </ sub >| yields a horizontal force to be:
Substituting the derivative of u < sub > ρ </ sub > into the expression for velocity:
Substituting the derivatives of u < sub > ρ </ sub > and u < sub > θ </ sub >, the acceleration of the particle is:
Substituting the data ( from Wikipedia ) one will get − 0. 7395 ° day < sup >− 1 </ sup >, a value substantially different from zero!
Substituting for F, we have P = 2ρQ ( V < sub > i </ sub > − u ) u.
Substituting these in gives the value of about 2. 7 × 10 < sup >− 14 </ sup > m. ( The true radius is about 7. 3 × 10 < sup >− 15 </ sup > m .) The true radius of the nucleus is not recovered in these experiments because the alphas do not have enough energy to penetrate to more than 27 fm of the nuclear center, as noted, when the actual radius of gold is 7. 3 fm.
Substituting cos θ < sub > c </ sub > for sin θ < sub > r </ sub > in Snell's law we get:
Substituting D from this equation into the previous and solving for Δ < var > t '</ var > gives:
Substituting p = ( μ + 1 )< sup >− 1 </ sup > gives g '( t ) = (( μ < sup >− 1 </ sup > + 1 )· e < sup >− t </ sup > − 1 )< sup >− 1 </ sup > and κ < sub > 1 </ sub > = μ.
Substituting p = μ · n < sup >− 1 </ sup > gives g '( t ) = (( μ < sup >− 1 </ sup > − n < sup >− 1 </ sup >)· e < sup >− t </ sup > + n < sup >− 1 </ sup >)< sup >− 1 </ sup > and κ < sub > 1 </ sub > = μ.
Substituting p = ( μ · n < sup >− 1 </ sup >+ 1 )< sup >− 1 </ sup > gives g '( t ) = (( μ < sup >− 1 </ sup >+ n < sup >− 1 </ sup >)· e < sup >− t </ sup >− n < sup >− 1 </ sup >)< sup >− 1 </ sup > and κ < sub > 1 </ sub > = μ.
Substituting and sub
Substituting for V '< sub > in </ sub > in the first expression,
Substituting and 1
Substituting this expression and ( 1 ) in ( 3 ) gives
Substituting this into ( 1 ) above, the period T of a rigid-body compound pendulum
Substituting and into
Substituting this into ( 2 ) gives
Substituting into the previous expression for,
Substituting this into the definition of the derivative via a difference quotient and taking limits means that the higher order terms – and higher – become negligible, and yields the formula interpreted as
Substituting into the prior equation, we have
Substituting this into the gain equation, we obtain:
Substituting this into the gain equation and solving for:
Substituting these two back into the Euler-Lagrange equation () results in
Substituting into the Binet equation yields the equation of trajectory
Substituting this definition into the prior equation, we find the general gain equation:
Substituting both quantities into the above equation generates the formula:
Substituting this expression into the first equation gives after some rearrangement
Substituting into the quadratic form gives an unconstrained minimization problem:
Substituting this solution into the wave equation above yields the paraxial approximation to the wave equation:
Substituting these values into the above equation gives T = 2π.
Substituting the above 2 results ( for E and J respectively ) into the continuum form shown at the beginning of this section:
Substituting energy into the equation gives:
Substituting equation into equation gives, which can be substituted into equation so that:
Substituting Equation ( Q ) into Equation ( I ) and then Equation ( I ) into Equation ( V ) gives:
Substituting into the equation for the change in entropy:
Substituting and identity
Substituting y = π gives the identity
Substituting y < sub > p </ sub > into the differential equation, we have the identity
Substituting and removing
* " Substituting A for B " when the intended meaning is " substituting B for A " or " replacing A with B ", i. e. " removing A and putting B in its place.
Substituting and common
Substituting one obtains another common form for the series, as
Substituting and gives
Substituting and evaluating the time derivative gives
Substituting for the masses in the equation for the Roche limit, and cancelling out gives
Substituting the coordinates above and rearranging the equation gives
Substituting and cancelling minus signs gives:
Substituting into equation ( iii ) and solving for y gives this value of y:
Substituting Ohm's law for conductances gives
Substituting Ohm's law for conductance then gives,
Substituting these in for P gives
Substituting propofol for midazolam, which gives the patient quicker recovery, is gaining wider use, but requires closer monitoring of respiration.
Substituting it into the KdV equation gives the ordinary differential equation
0.231 seconds.