Mixed second partial

In this case, y is held fixed in both differentiations. In addition, there are mixed second partial derivatives, such as the derivative with respect to x and then with respect to y ... 

The Euler reciprocity relation is an identity relating the mixed second partial derivatives. If z = z(x, y) is a differentiable function, then the two different mixed second partial derivatives must equal each other ... 

Hence, it is acceptable to reverse the order of mixed second partial differentiation of an exact differential without affecting the final result. 

However complicated the fluid dynamics problem might be, each component of the velocity vector can be solved analytically or numerically. Hence, each u, exists, it is unique and it is an exact differential. This means that the order of mixed second partial differentiation of each n, can be reversed without affecting the final resnlt. Now... 

Vorticity Equation for Creeping Flow of an Incompressible Newtonian Fluid. Since all scalar components of the velocity vector are exact differentials, it is permissible to reverse the order of mixed second partial differentiation without affecting the final result. If this procedure is performed twice, then inspection of summation representations of the following two vector-tensor operations reveals that they are equivalent ... 

Step 3 is satisfied because P is an exact differential, which implies that the order of mixed second partial differentiation can be reversed without affecting the final result. It might be worthwhile to verify this test for exact differentials, given by (8-139), before embarking on the integrations in steps 1 and 2. 

These coefficients are themselves, in general, functions of the independent variables and may be differentiated to give mixed second partial derivatives for example ... 

The following is called a mixed second partial derivative ... 

Euler s reciprocity relation states that the two mixed second partial derivatives of a differentiable function must be equal to each other ... 

We refer to the second partial derivatives in Eqs. (B-12a) and (B-12b) as mixed second partial derivatives. The Euler reciprocity relation is a theorem of mathematics If / is differentiable, then the two mixed second partial derivatives in Eq. (B-12a) and (B-12b) are the same function ... 

For a function of three variables, there are nine second partial derivatives, six of which are mixed derivatives. The mixed second partial derivatives obey relations exactly analogous to Eq. (B-13). For example. 

Last but not least, suppose / (xj,..., x ) exists such that / and its first and second partial derivatives are continuous in the neighborhood of a point r = ( Ti,..., Xk) G K Under this presupposition, the theorem of Schwarz holds, which states that for the mixed second-order partial derivatives the order of differentiation is irrelevant that is. 

The second partial derivative d //dydx, for instance, is the partial derivative with respect to y of the partial derivative of / with respect to x. It is a theorem of calculus that if a function / is single valued and has continuous derivatives, the order of differentiation in a mixed derivative is immaterial. Therefore the mixed derivatives jdydx and d f /dxdy,... 

The time rate of change of shear strain, y (or dyidt, the dot is Newton s notation for the time derivative), is the so-called shear rate. Since the order in which the mixed second derivative is taken is immaterial (note that by sticking to two dimensions, we can write total rather than partial derivatives), then... 

For a quaternary Ai xBxCyDi y alloy, the stability criterion is determined by the second partial derivative of the free energy change, AG, for mixing ... 

The second variation was prepared with a similar method as the first variation. However, after mixing the partial solution of ethanol (20 g) glycerol (4 g), zein (2g ) and varying amounts of nanoclay (0.5g and Ig). The solution was treated for 90 seconds with ultrasonic energy. The horn used was a 10 mm flat tip catenoidal horn and the tip amplitude was approximately 30 pnipp. A Branson Sonifier was used for ultrasonic treatment. After treatment the balance of zein (8 g) and ethanol (20 g) was added. This procedure was followed to maintain a relatively low viscosity suspension to enhance ultrasonic treatment. 

Urea.—Forma.IdehydeResins. Cellular urea—formaldehyde resins can be prepared in the following manner an aqueous solution containing surfactant and catalyst is made into a low density, fine-celled foam by dispersing air into it mechanically. A second aqueous solution consisting of partially cured urea—formaldehyde resin is then mixed into the foam by mechanical agitation. The catalyst in the initial foam causes the dispersed resin to cure in the cellular state. The resultant hardened foam is dried at elevated temperatures. Densities as low as 8 kg/m can be obtained by this method (117). 

It is well known, that in aqueous solutions the water molecules, which are in the inner coordination sphere of the complex, quench the lanthanide (Ln) luminescence in result of vibrations of the OH-groups (OH-oscillators). The use of D O instead of H O, the freezing of solution as well as the introduction of a second ligand to obtain a mixed-ligand complex leads to either partial or complete elimination of the H O influence. The same effect may be achieved by water molecules replacement from the inner and outer coordination sphere at the addition of organic solvents or when the molecule of Ln complex is introduced into the micelle of the surfactant. 

In the two-stage operation, the feed is hydrodesulfurized in the first reactor with partial hydrocracking. Reactor effluent goes to a high-pressure separator to separate the hydrogen-rich gas, which is recycled and mixed with the fresh feed. The liquid portion from the separator is fractionated, and the bottoms of the fractionator are sent to the second stage reactor. 

A schematic flow diagram of this process is presented in Figure 12 which depicts three catalyst beds operating in series. Fresh make-up gas is mixed with total recycle and fed to the first bed. The effluent from the first bed is partially cooled so that when it is combined with another portion of cold make-up, the mixture is at or above the initiating reaction temperature. The mixture is fed to the second bed. This procedure is continued to the third and subsequent beds. 

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