Page 73 - Introduction to Continuum Mechanics

58 Curvilinear Coordinates

The geometrical representation of de r and d*Q are shown in the following figure where one
notes that e^P) has rotated an infinitesimal angle d6 to become e/CO^erCO+^r where de r is
perpendicular to e^P) with a magnitude \de r\=(l)(dB}. Similarly de0 is perpendicular to
e@(P) but is pointing in the negative e r direction and its magnitude is also (l}dO,

From the position vector r=re n we have

Using Eq. (2D1.2a), we get

The geometrical representation of this equation is also easily seen if one notes that dris the
vector PQ in Fig. 2D.2. The components of V/, Vv etc. in polar coordinates will now be
obtained.

Fig. 2D.2

(i) Components ofVf
Let/(r,0) be a scalar field. By definition of the gradient of/, we have
df= V/-rfr= [afr+a&e]].[dnr+nl09o]

where a r and OQ are components of Vf in the e r and eg direction respectively.
Thus,