Yajnik and Lieber 



We will briefly depart from the vector -diadic notation to introduce a tensor 

 of third rank. Rewriting Eqs. (5) and (6) in the tensor notation, we have 



^ = ^ijc "t,k tj t^ = e.., v,^ t.t^ (5a) 



where - ' " 



^ij = ^i,j - "k,k h/3. (6a) 



To exploit the symmetry in t . tj^ , we define 



^i:jk = (^ij£ "t.k + ^ikS "t,j)/2 



= (^ii8 v,k + ^ike ^ej)'2 C^) 



SO that 



Wi = Ri:,U tjt, (8) 



Each component of the angular velocity of the line element is thus a quadratic 

 form in its direction cosines, and the coefficients of the quadratic form consti- 

 tute a tensor of third rank. The tensor R^ . ^ is symmetric in j and k and will 

 be called the rotation tensor. Note that the rotation tensor and hence the angu- 

 lar velocity are independent of dilation V-u . 



To analyze the rotation of a material surface, let its location at time t be 

 given by f(x, t) = . Note that f will be zero on the surface. If n is a unit 

 normal at a given particle, 



R = \ V f, H-H = 1, H-H = ' (9) 



for some k , and . , 



H=\Vf+\Vf-(vri)- n. (10) 



Defining the angular velocity w of the normal as 



w = n X n, (11) 



we get 



J. _* _ 



n = w X n, 



and 



w = [(V u)- n] X n = (n-V) x n. (12) 



Note the conjugate character of Eqs. (5) and (12). Rewriting Eq. (12) below in 

 tensor notation, we have 



^eji "k,i! "j"k - ^eji "ke "j"k. ' ^^ 



n- n, = e„ ■ • v, „ n-n,. 



We define 



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