00060___a5004ae1ea0940be8fb246299dbaa8fa

00060___a5004ae1ea0940be8fb246299dbaa8fa - potential. 2.5....

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Sec. 2.5 TENSOR FIELDS OF HIGHER RANK 39 A contravariant vector is indicated by a superscript, called a contravariant index. A differential doi is a prototype of a contravariant vector: (3) -. ap doa = - dd" ad'" Hence, a vector is contravariant if it transforms like a differential.+ A system is called a covariant vector field, or a covariant tensor field of rank one, if it has three components r]i in the variables O', 02, O3 and three components fji in the variables g', 82, e3, and if the components in these two coordinate systems are related by the law A covariant vector is indicated by a subscript, called a covariant index. A gradient of a scalar potential qtJ transforms like (4), (5) Hence, a vector is covariant if it transforms like a gradient of a scalar
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Unformatted text preview: potential. 2.5. TENSOR FIELDS OF HIGHER RANK A system such as the three-dimensional Euclidean metric tensor gij has nine components when i and j range over 1, 2, 3. Such quantities are given special names when their components in any two coordinate systems are related by specific transformation laws. Covariant tensor field of rank two, t i j Contravariant tensor field of rank two, tij 'The variables Bi and 8", in general, are not so related [see Eq. (2.3:9)]. Thus, although the differential doi is a contravariant vector, the set of variables Bi itself does not trans- form like a vector. Hence, in this instance, the position of the index of Bi must be regarded as without significance....
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This note was uploaded on 01/04/2012 for the course ENG 501 taught by Professor Thomson during the Fall '05 term at MIT.

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