129
VECTOR ANALYSIS
CHAP. 5]
—
3.
=
4.
curl A
—
Laplacian of $ =
These reduce to the usual expressions in rectangular coordinates if we replace (ui, u2, Us) by (x,y,z), in which case ei, e2 and e3 are replaced by i, j and k and hi-h2 = hs = 1. SPECIAL CURVILINEAR COORDINATES 1. Cylindrical Coordinates (p, <j>, z). See Fig. 5-10. Transformation equations: x — p cos <t>, y = p sin <£, z = z where p è 0, 0 ë <j> < 2*, -°° < z < «>. Scaie factors: hi = 1, hz = p, ha = 1 Element of arc length: ds2 = dp2 + p2 cZ<£2 + dz2 /aco&wm: Element of volume: dV — pdpd<j> dz Laplacian:
Fig.5-10
Note that corresponding results can be obtained for polar coordinates in the plane by omitting z dependence. In such case for example, ds2 = dp2 + p2 d<j>2, while the element of volume is replaced by the element of area. dA = p dp d<j>. 2. Spherical Coordinates (r,0,<f>). See Fig. 5-11. Transformation equations: x = r sin 0 cos ó, y = r sin 6 sin A, z — r cos 6 where Scale factors: Element of arc length: Jacobian: Element of volume: Laplacian Other types of coordinate systems are possible.
Fig.5-11