Leibnitz, rule for differentiating an integral

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... 

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. 

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See also in sourсe #XX -- [ ]

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