Note: The above operations may be generalized to cylindrical coordinates by replacing (x, y, z ) by (r, 6, z), and to spherical coordinates by replacing (x, y, z by ( r , 6, 4). ) Descriptions of curvilinear coordinates are given in Figures 1.2-2, A.6-1, A.8-1, and A.8-2.
**.DIFFERENTIAL OPERATIONS FOR SCALARS, VECTORS, AND TENSORS IN CARTESIAN COORDINATES
x v],= - - dy dz dv, dz
dv, [ V x v ] =--Y dz
dvZ dx [V
dvy ax - dux aY
dv, dvy (V.v)=-+-+dx dy
d2vz d2v, d2vZ +-az2 = [V2v], [V Vv],= ax2 dvx dvx dvx [v Vv],= vx - + vY - + v, -dx dy dz
[v' Vv],= vx- + v
~(v,v,) a(vyvx) d(v,vX) [V vv], = - ------ + dx + dy dz a(vXvy) a(vYvy) ~(v,v,) [V .vv],= - dy dx +-+dz
[V vv],= ---- +-+-
a(vXvz) d(vyvz) ~(v,v,) dx dy dz
(T :V v ) =
dvx dux dux rxx- + r - + rxzdx dy dz
Note: the differential operations may not be simply generalized to curvilinear coordinates; see Tables A.7-2 and A.7-3.
+ v, dvz dz
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R. Byron Bird Warren E. Stewart Edwin N. Lightfoot
Chemical Engineering Department University of Wisconsin-Madison
John Wiley & Sons, Inc.
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