00069___295fcc0c55805bdcd422f01f5b88030e - since the topic...

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48 TENSOR ANALYSIS Chap. 2 2.12. COVARIANT DIFFERENTIATION OF VECTOR FIELDS The generalization of the concept of partial derivatives to the concept of covariant derivative, so that the covariant derivative of a tensor field is another tensor field, is the most important milestone in the development of tensor calculus. It is natural to search for such an extension in the form of a correction term that depends on the vector itself. Thus, if [a is a vector, we might expect the combination to be a tensor. Here the suggested correction is linear function of ti, and F(i,j, a) is some function with three indices. The success of this scheme hinges on the Euclidean Christoffel symbols, which are certain linear combi- nations of the derivatives of the metric tensor gij. This subject is beautiful, and the results are powerful in handling curvilinear coordinates. However,
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Unformatted text preview: since the topic is not absolutely necessary for the development of solid me- chanics, we shall not discuss it in detail, but merely outline below some of the salient results. We discussed in Sec. 2.3 the metric tensor gij in a set of general coordi- nates (xl, x2, z3), and in Sec. 2.6 the associated metric tensor gij. By means of these metric tensors, the Euclidean Christoflel symbols r&(xl, x2, x3) are defined as follows: The Ei = f'(xl, x2, x3) as follows (see Prob. 2.24, p. 55) is not a tensor. It transforms under a coordinate transformation ?P This equation can be solved for d2xX/b%"b%@ by multiplying (2) with bxm/aei and sum over i to obtain (3) Interchanging the roles of xi and Zi and with suitable changes in indices, we can substitute (3) into Eq. (2.11:5) to obtain axx a x p 1 ' a? - ap axp aei aei _ . b P den - r;&)-- aee axPbe:"bxX...
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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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