5. The Vector Product. Curl of a Vector

5. The Vector Product. Curl of a Vector The tensor e defined in Sect. 4 can be used in the definition of the well known vector prodztct of two vectors A and B in a coordinate manifold of three dimensions. According as the vectors A and B are covariant or contravariant vectors we define their vector product to be the vector C having components represented by one or the other of the following two sets of equations

Cs = eiik Ai Bk;

Ci= eEikAi Bk.

(5.1)

As defined by the equations (5.1) the vector product of two absolute covariant vectors A and B is a relative contravariant vector C of weight + 1 while the vector product of two absolute contravariant vectors A and B is a relative covariant vector C of weight -1 in the three dimensional coordinate manifold. An analogous application of the tensor e furnishes the definition of the curl of a vector V in a coordinate manifold of three dimensions. Thus if V is an absolute covariant vector whose components V i ( x ) are differentiable functions of the coordinates x1,x2,x3 of the allowable coordinate systems covering the manifold we see immediately from the equations of transformation of the components of V that the quantities

are the components of a covariant tensor of the second rank. Hence the quantities W idefined by

wi=

+

1 in the are the components of a relative vector W of weight three dimensional manifold. The vector W defined by ( 5 . 2 ) is called the curl or rotation of the vector V . 16