Leibnitz s rule

Differentiating equation (II-8) with respect to time and using Leibnitz s rule, one can obtain the evolution of P(t) with time. A Laplace transform analysis will finally yield (58,59) ... 

This completes the list of the mathematical tools required to reformulate the volume averaged equation (3.121) into a more practically useful form. Applying the Leibnitz s rule (3.123) to the first term in (3.121) yields ... 

The first, second and third terms in (3.218) have to be reformulated using the conventional time averaging theorems . The first theorem one makes use of relates the average of a time derivative to the time derivative of an average quantity, and is called the Leibnitz s rule for time averaging ... 

The extended form of the Leibnitz s rule can be derived by use of the phase indicator function (3.260), the topological equation (3.269), the Reynolds averaging rules (1.379) and the chain rule [54] ... 

In accordance with the conventional Re3molds axioms (1.379), the Leibnitz s rule is given by ... 

By Leibnitz s rule, the change in the area of the stable aerosol with time is... 

While we might integrate this with our chosen relaxation function, it is much easier to simply differentiate the integral directly, using Leibnitz s rule, to get the modulus. This is not difficult because all time dependence can be factored out of the integral. Putting this all into place we have finally ... 

Using Leibnitz s rule to differentiate this integral, keeping in mind that the stress is a function of the dummy variable and not the limit Xl gives directly... 

Based on the defining equation for (f"), all the functional dependence of on t" appears in the upper limit of integration. Therefore, only the first term on the right side of (11-215) (i.e., Leibnitz s rule) is required to calculate d /dt", after one associates surface area S with the generic integrand g. The result is... 

The second equation is the same as Leibnitz s rule for the differential operator. The coordinate arguments of the Ursell functions have been suppressed in Eq. (86). 

Applying Leibnitz s rule and using the fact that ... 

Rearranging and differentiating using Leibnitz s rule gives... 

Israelachvili (2011) describes how this definition for a force law arose historically. Equation 10.2 arises from Equation 10.1 by using Leibnitz s rule of differentiation of integrals. Negative F and V imply attraction while positive F and V mean repulsion. 

Applying the Leibnitz s rule (3.128) to the first term in (3.126) yields ... 

The first term on the RHS gives the change in the integral because the function itself is changing with time, the second term accounts for the gain in area as the upper limit is moved in the positive axis direction, and the third term accounts for the loss in area as the lower limit is moved. The formal derivation of the Leibnitz s rule can be found elsewhere [8, 45],... 

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