Page 122 - Introduction to Continuum Mechanics

Time Rate of Change of a Material Element 107

Taking the material derivative of Eq. (i), we obtain

Now,

d (*»0
where v(X^) an v are tne material and the spatial description of the velocity of the
particle X, therefore Eq. (ii) becomes

Thus, from the definition (see Section 2C3.1) of the gradient of a vector function, we have

and

/** /*s.
In Eq. (3.12.2) the subscript X in (V xv) emphasizes that (V xv) is the gradient of the material
description of the velocity field v and in Eq. (3.12.3) the subscript x in (V xv) emphasizes that
(V xv) is the gradient of the spatial description of v.
In the following, the spatial description of the velqejty function will be used exclusively so
that the notation (Vv) will be understood to mean (V xv). Thus we write Eq. (3.12.3) simply as

With respect to rectangular Cartesian coordinates, the components of (Vv) are given by