Order of differentiation

Order of Dijferentiation It is generally true that the order of differentiation is immaterial for any number of differentiations or variables provided the function and the appropriate derivatives are continuous. For z =f x, y) it follows ... 

Modeling of Chemioal Kinetios and Reaotor Design Since the order of differentiating is indifferent, then... 

For well-behavedb functions, the order of differentiation does not matter, so... 

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... 

The governing partial differential equation for G(t,z) is obtained by differentiating both sides of Equation 10.1 with respect to k and reversing the order of differentiation. The resulting PDE for Gj,(t,z) is given by (Seinfeld and Lapidus, 1974),... 

The change in the order of differentiation to give the last term in equation A.29 is permissible because the velocity field will satisfy the sufficient conditions, namely that the two mixed partial derivatives are continuous. Equation A.27 can now be written as... 

The second term on the right hand side of Equation 3.16 introduces complications because it couples x[ and xj. The first term, on the other hand is easily solved because it involves no coupling. The resolution of the difficulty introduced by the second term is to take advantage of the symmetry of the fy matrix. Note that each f is an element of a symmetric matrix and the second derivatives fy are independent of the order of differentiation. There is a well known mathematical theorem on the diagonalization of symmetric matrices which states (as applied to Equation 3.16) that when we introduce a new coordinate Q ... 

But notice that the right-hand side of Eq. (5.19) is the value of the tangent that also goes through point A therefore, the tangency point along the Hugoniot curve is J. Since the order of differentiation on the left-hand side of Eq. (5.19) can be reversed, it is obvious that Eq. (5.15) has been developed. 

That is, the order of differentiation is immaterial for any function of two variables. Therefore, if dL is exact. Equation (2.23) is correct [8]. 

The Jacobian matrix defined in (3.41) can be easily computed by the same interpolation technique. The idea is to differentiate (3.60) with respect to the parameters changing the order of differentiation and spline integration. 

Since the order of differentiation is immaterial for exact differentials, it follows that dldA[(dGldp)A]p = dldp[(dGldA)p]A or... 

Because the axial-momentum equation requires that dp/dz = 0, and the order of differentiation can be exchanged, the pressure term must vanish. As noted in Bird et al. [35], rearranging the equations leads to the following observation ... 

The l.h.s. of Eq. (2.38) is the /th element of the vector g(4+I). On the r.h.s. of Eq. (2.38), since the partial derivative of q with respect to its / th coordinate is simply the unit vector in the /th coordinate direction, the various matrix multiplications simply produce the / th element of the multiplied vectors. Because mixed partial derivative values are independent of the order of differentiation, the Hessian matrix is Hermitian, and we may simplify Eq. (2.38) as... 

The results of two consecutive differentiations of the A-matrix with respect to two spatial coordinates, p and q, does not depend on the order of differentiation [20]. This requirement leads to a set of conditions similar to equation (19) (or (20)) but where t is replaced by t (see equation (42)). 

Interchanging the order of differentiation and introducing the sensitivities yields... 

But by definition, we have selected a fixed control volume, therefore the order of differentiation by time and integration can be reversed to get ... 

The first relation follows directly by taking the partial derivative of G = H — TS with respect to nt (holding P, T, and the other n- s constant). The latter two relations can be proved by reversing the order of differentiating with respect to , and with T and P respectively. 

Field Mathematical representation of a physical quantity at every point of space the mathematical quantity is defined as continuous for all necessary orders of differentiation. 

Since the order of differentiation in mixed second derivatives is immaterial, these equations give... 

Since the order of differentiation is immaterial, we can find a relation between any two of the functions listed in (1.11.9) according to the following procedure ... 

The restrictions on the summations in Eqn. (66) are the same as those in Eqn. (60). This result can also be obtained from the idempotency of the one-electron density, which is usable through all orders of differentiation too [62,63]. Equation (65) is used for elements of U that are in diagonal blocks. One may freely choose those blocks to be symmetric and then the elements are given by... 

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