Implicit differentiation

It is worthwhile, albeit tedious, to work out the condition that must satisfied in order for equation (A1.1.117) to hold true. Expanding the trial fiinction according to equation (A1.1.113). assuming that the basis frmctions and expansion coefficients are real and making use of the teclmiqiie of implicit differentiation, one finds... 

I hese equations cannot be used directly, and numerical methods are needed to compute the velocity components. The velocity components can be found by implicit differentiation and using an iterative technique.-" ... 

Reversing the order of differentiation on the left-hand side of Equation 6.7 and performing the implicit differentiation of the right-hand side, we obtain... 

When the output vector (measured variables) are related to the state variables (and possibly to the parameters) through a nonlinear relationship of the form y(t) = h(x(t),k), we need to make some additional minor modifications. The sensitivity of the output vector to the parameters can be obtained by performing the implicit differentiation to yield ... 

The relationship between (Sy/Sk,) and G is obtained by implicit differentiation of Equations 10.4 - 10.7 depending on the type of measurements we have. Namely,... 

The overall sensitivity of the well production rate can now be estimated through implicit differentiation from the sensitivity coefficients already computed as part of the Gauss-Newton method computations as follows... 

Let us implicitly differentiate both sides of (6.27) with respect to A, using the general chain rule to obtain... 

The second term on the RHS of eq. 22 is the volume expansivity which can be calculated from a two-parameter, cubic equation of state such as the Redlich-Kwong EOS.(14) We chose the Redlich-Kwong EOS because the "a" term is independent of temperature, which simplifies the procedure of solving for the analytical solution. Using the method of implicit differentiation with the Redlich-Kwong equation of state,... 

This relation can be used to relate changes in wave energy density to changes in wave velocity in a lossless medium, i.e., one in which P is constant Implicitly differentiating Equation 2.47 yields... 

An implicit differential equation is an equation in which the dependent variable is expressed as a function of its derivative ... 

Petzold has provided a numerical package, dassl, to compute solutions to implicit differential, and differential-algebraic equations. The main difference between using dassl and 1 sode is the form of the user-supplied function defining the model A second difference is that the user must supply xo as well as xo- The reader can consult the software for Example 8.1 at www.che,wisc.edu/ jbraw/chemreacfun to see the details of how to use dassl. 

The Joule-Thomson coefficient juj can be related to other thermodynamic derivatives by application of the formula of implicit differentiation [Eq. (A-10)]. Thus we may write... 

Equation (A-9) is the equation for implicit differentiation. For three variables, z = z x, y), it has the more familiar form... 

Implicit differentiation is a method of finding dytdx for a functional relation that cannot be easily solved for y(x). Suppose we have y and x related by... 

Implicit differentiation can be applied more generally to any functional relation of the form f(x, y) = 0. 

The components of the Jacobian matrix can be obtained using implicit differentiation, they are ... 

Note that the objective function is concave, and hence the retailer s first-order condition characterizes the unique best response. For stationary policies, it is sufficient to consider the best response functions in the single-period game. The slope of the retailer s best-response function is found by implicit differentiation as follows ... 

Diva [4] is a dynamic simulator for chemical plants. The model of a plant is expressed in a linearly implicit differential-algebraic model formulation... 

The polynomial format of the Bender multiparameter EoS yields a simple solution via a Newton-Raphson iteration procedure [9]. We found three iterations provided fluid-phase density values with an associated RCSU of < 2 x 10" % between the third and fourth iteration. We took the value of the 6 -iteration, which gave an RCSU of 0.0% between the 4 - and 6 -iteration. The form of the Bender EoS requires implicit differential expressions [10] to evaluate and propagate the uncertainty to generate the CSU in the amoimt of fluid change from the material balance, leading to u n). Implicit differentials were developed for... 

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