By Holt Rinehart & Winston
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Additional resources for Weather and Climate Grade 7 (Holt Science & Technology Modules 2005)
41) which is known as the Gram determinant. The proof, however, will be given later. 42) which involves all elements gij of the metric tensor. 43) so that the scalar triple product involving the general basis vectors q1 , q2 , q3 can easily be evaluated. This will be done in some detail when we consider various √ coordinate systems. 43), g is called the functional determinant of the system. 44) which is also known as the Grassmann rule. 44) is not unique. The proof of this equation will be given later with the help of the so-called reciprocal coordinate system.
The height h of the parallelepiped is found by projecting the vector A onto the cross product B × C. If the volume vanishes then the three vectors are coplanar. This situation will occur whenever a vector appears twice in the scalar triple product. It is apparent that, in the scalar triple product, any cyclic permutation of the factors leaves the value of the scalar triple product unchanged. 37) It is regrettable that the symbol g is also assigned to the acceleration due to gravity, but confusion is unlikely to occur.
56) is equal to 1. 57) are inverse relations. Often it is desirable to work with unit vectors having the same directions as the selected three linearly independent basis vectors. 4 Reciprocal coordinate systems 17 Owing to the symmetry relations gij = gj i and g ij = g j i each metric tensor is completely speciﬁed by six elements. 13) of the scalar product. 61) While in the Cartesian coordinate system the metric fundamental quantities are either 0 or 1, we cannot give any information about the gij or g ij unless the coordinate system is speciﬁed.