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By Campbell J.E.

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Extra resources for A Course of Differential Geometry

Sample text

The product two a tensor whose components are the products of each component of the first and each component of the second tensor. The upper integers of the of tensors is product are the upper integers of the two factors, and the lower integers of the product are the lower integers of the two Two factors. tensors of the same character that is, with the same number can be of each kind of integers, upper and lower which have the if we the take together added, components same we They can integers.

I integers dummies, and we should avoid the use of 71 as a are not dummy. /e 1 a ( e^ 5 ) 5 By aid of the symbolism thus introduced we can avoid a prolixity which would otherwise almost bar progress. A very little practice will enable one to use this symbolism freely, 6. We and when necessary to express the results explicitly. 1) ; and therefore the operator p annihilates each of the functions a ik which, of course, could have been written Tik We have a a^ = . , ; and therefore /fc u u r a*7 + a (tpk) + a (kpt) dx p = 0.

It should be rethat the symbol on the right stands for the sum of 1, 2, ... a. symbols, with p membered n ] Tensors and tensor components defined. We must is meant what a tensor. have We seen how explain by functions denoted by 7. now r may be derived from with respect to X1 ... a > (iy functions xn The . Py * 6. * * ty $,, ... and their derivatives different functions obtained allowing the integers to take called components of the set. values from all Suppose that we transform to ne\v variables 1 by up to n are a;'.

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