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58 CURVILINEAR COORDINATE SYSTEMS
or using the Levi-Civita symbol and subscript notation,
3.5 THE GRADIENT, DIVERGENCE, AND CURL IN CYLINDRICAL
AND SPHERICAL SYSTEMS
3.5.1 Cylindrical Operations
In the cylindrical system h, = h, = 1, h2 = hg = p , and h3 = h, = I. The gradient, divergence, and curl become
3.5.2 The Spherical Operations
In the spherical system hl = h, = I, h2 = he = r , and h3 = h+ = r sin 0. The
gradient, divergence, and curl become
- dQt 1 dQt 1 dQ, VQ, = - 4 , + --qe + ~- dr r 30 r sin 0 a+ ‘ 4 1 dA+ +-- (3.67)
- _ I d ( r 2 A , ) I d(sin0Ae) V. A = - - +- r 2 dr r sin 0 d0 r sin 0 J+
EXERCISES^59
EXERCISES FOR CHAPTER 3
1. A vector expressed in a Cartesian system has the following form:
v = Cx + C, + C Z.
(a) If this vector exists at the Cartesian point (1,2, l), what are its components
(b) If this vector exists at the Cartesian point (1,2, l), what are its components
(c) If this vector exists at the Cartesian point (0, 0, l), what are its components
(d) If this vector exists at the Cartesian point (0, 0, l), what are its components
- In a cylindrical system, with coordinates ( p , 4, z ) , the vector Vl exists at the point (1,0,O) and is given by = C, + 6,. A second vector '5, exists in the same coordinate system at the point (117r/2, 0) and is given by v 2 = CP + 64. What is the value of V,. V,?
expressed in a cylindrical system?
expressed in a spherical system?
expressed in a cylindrical system?
expressed in a spherical system?
- A circular disk, centered at the origin of a coordinate system, is rotating at a constant angular velocity 0,.
(a) In Cartesian coordinates, what is the velocity vector of the point P located
(b) In polar coordinates, what is the velocity vector of the point P located at the
at the Cartesian coordinates (1, l)?
Cartesian coordinates (1, l)?
4. Perform the integral
where
F = 2 sin 0 sin 4 6, + 3 cos 0 cos 4 C+ + 4 cos 0 cos 4 60,
