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EXERCISES 43
(d) This is a conservative field. What is its potential? (e) Sketch the equipotentials.
- The continuity equation for a fluid is
JP at
L -
V. J + - = O ,
where J = p v. (a) Show V - V = 0 if the fluid is incompressible (constant density). (b) Apply Gauss’s Theorem to the continuity equation and interpret the result.
- Prove these integral forms of the differential operators:
13. Maxwell’s equations in vacuum are
Manipulatethese using subscript/summationnotation to obtain the wave equation in vacuum:
CURVILINEAR COORDINATE SYSTEMS
Up to this point, our discussions of vector, differential, and integral operations have been limited to Cartesian coordinate systems. While conceptually simple, these sys- tems often fail to utilize the natural symmetry of certain problems. Consider the electric field vector created by a point charge q, located at the origin of a Cartesian system. Using Cartesian basis vectors, this field is
(3.1)
In contrast, a spherical system, described by the coordinates ( r , 8, +), fully exploits the symmetry of this field and simplifiesEquation 3.1 to
The spherical system belongs to the class of curvilinear coordinate systems. Basis vectors of a curvilinearsystem are orthonormal,just like those of Cartesian systems, but their directions can be functions of position. This chapter generalizes the concepts of the previous chapters to include curvi-
linear coordinate systems. The two most common systems, spherical and cylindrical,
are described first, in order to provide a framework for the more abstract discussion of generalized curvilinear coordinates that follows.
3.1 THE POSITION VECTOR
The position vector F(P) associated with a point P describes the offset of P from the origin of the coordinate system. It has a magnitude equal to the distance from the origin to P, and a direction that points from the origin to P.
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