Leibnitz rule

Noting that R is the upper limit of the integral, and using the Leibnitz rule ... 

This can be solved for the shear rate at the tube wall (yw) by first differentiating Eq. (6-92) with respect to the parameter rw by application of Leibnitz rule to give... 

The composition gradient is obtained by differentiating Eq. 10.3.8 (using the Leibnitz rule for the derivative of an integral)... 

To proceed we need to apply some particular forms of the Leibnitz rule and of the Gauss theorems. 

Fig. 1.4. Sketch of the control volume determining the basis for area-averaging of the single phase equations. For the mathematical derivation of the limiting form of the Leibnitz rule and of the Gauss theorem, note that n T n ij and Az = As cos 6 = As(n.43 0...

The first, second and third terms in (3.122) have to be reformulated using the conventional volume averaging theorems . The first theorem one makes use of relates the spatial average of a time derivative to the time derivative of a spatial average, and is called the Leibnitz rule for volume averaging ... 

In this theorem A/ is the area of the interface between phase k and the other phase, Ilk is the outward unit normal of the infinitesimal element of area a of phase k, and v/ is the velocity of the local interface. The theorem, which was originally derived by [236], represents a special form of the Leibnitz rule which is necessary for the particular case when the time derivative is discontinuous and reflects a Dirac delta function like character [90, 239, 58]. 

A three dimensional extension of the Leibnitz rule for differentiating an integral is relevant for the derivation of the governing transport equations L In the material (Lagrangian) representation of continuum mechanics a representative particle of the continuum occupies a point in the initial configuration of the continuum at time t = 0 and has the position vector = (Ci, 2, Cs)-In this -space the coordinates are called the material coordinates. In the Eulerian representation the particle position vector in r-space is defined by r = (ri,r2,r3). The coordinates ri,r2,r3 which gives the current position of the particle are called the spatial coordinates. Let be any scalar, vector... 

This is essentially a generalization of Leibnitz rule for differentiation of a one-dimensional integral with respect to some variable when both the integrand and the limits of integration depend on that variable. The proof of (2-10) is straightforward.7 We first note that every point x(f) within a material control volume is a material point whose position is prescribed by (2-9). Hence, once the (arbitrary) initial shape of the material control volume is chosen (so that all initial values of x0 are specified), a scalar quantity B associated with any point within the material control volume can be completely specified as a function of time only, that is, B [x(t), t]. Thus the usual definition of an ordinary time derivative can be applied to the left-hand side of (2-10), and we write... 

The derivative of erf(x) can be evaluated by the Leibnitz rule as described in the next section. 

The Leibnitz rule (7) furnishes a basis for differentiating a definite integral with respect to a parameter ... 

Step 4. The only dependence on y in the integral of (8-136) is found in the upper limit of integration, because yo is constant. Hence, the Leibnitz rule for differentiating a one-dimensional integral with variable limits yields ... 

Boundary condition (1) is employed for the lower integration limit in the following expression, and elements of the Leibnitz rule for differentiating an integral with variable upper limits are invoked to change the integration variable from to z. One obtains... 

Each of the three derivatives on the right side of this equation is evaluated separately, where the Leibnitz rule for differentiating an integral with variable limits is employed for (dP/d ) o- For example. 

Once again, the Leibnitz rule is useful, because if one defines... 

Step 1. Use the integral form of a linear-least squares analysis to determine the best value of the pseudo-first-order kinetic rate constant, i, that will linearize the reaction term in the mass transfer equation. It is necessary to apply the Leibnitz rule for differentiating a one-dimensional integral with constant limits to the following expression ... 

Applying the Leibnitz rule to the integral on the right-hand side gives ... 

Leibnitz rule If we have an integral of a continuous function / such as... 

Applying Leibnitz rule to differentiate both sides of (16.17) with respect to Tw gives... 

Parallel-plate torsional flow is a second choice. Assuming incompressible flow, the viscosity can be calculated from the total torque needed to turn one disk while keeping the other immobile. Following a derivation similar to that used for the Weissenberg-Rabinowitsch equation and using Leibnitz rule, it is straightforward to get the viscosity at the rim of the disk ... 

Using Leibnitz rule to differentiate Eq. (14.4.7) with respect to gives... 

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