00063___281c8411e6b293b798727511df8b3353 - Thus, tensor...

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42 TENSOR ANALYSIS Chap. 2 Using (3) and (4), we can write Since g is positive, Eq. (9) may be solved to give where Dij is the cofactor of the term gij in the determinant g. The tensor gij is called the associated metric tensor. It is as important as the metric tensor itself in the further development of tensor analysis. PROBLEM 2.10. Prove Eqs. (6), (7) and (10). Write down explicitly Di’ in Eq. (11) in terms of g11,glz,. . . ,g33. 2.7. THE SIGNIFICANCE OF TENSOR CHARACTERISTICS The importance of tensor analysis may be summarized in the following statement. The form of an equation can have general validity with respect to any frame of reference only if every term in the equation has the same tensor characteristics. If this condition is not satisfied, a simple change of the system of reference will destroy the form of the relationship. This form is, therefore, only fortuitous and accidental.
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Unformatted text preview: Thus, tensor analysis is as important as dimensional analysis in any formulation of physical relations. Dimensional analysis considers how a physical quantity changes with the particular choice of fundamental units. Two physical quantities cannot be equal unless they have the same dimensions; any physical equation cannot be correct unless it is invariant with respect to change of fundamental units. Whether a physical quantity should be a tensor or not is a decision for the physicist to make. Why is a force a vector, a stress tensor a tensor? Because we say so! It is our judgement that endowing tensorial character to these quantities is in harmony with the world. Once we decided upon the tensorial character of a physical quantity, we may take as the components of a tensor field in a given frame of reference...
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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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