By R. Courant, D. Hilbert
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London, 1959). An essay of exceptional value and a tremendous piece of historical research on the topic of the people who discovered the laws of planetary motion 4. R. Symon, Mechanics, 3rd edn. (Addison Wesley, Reading, Massachusetts, 1971). An excellent introductory text to classical mechanics 5. H. Goldstein, C. Poole, J. Safko, Classical Mechanics, 3rd edn. (Addison Wesley, New York, 2002). An advanced text, but pedagogically very clear 6. D. M. Lifshitz, Mechanics, 3rd edn. (Butterworth-Heinemann, Oxford, 2000).
Thus, A03 D R3j Aj means the sum over j , as j ranges over 1; 2; 3. (From now on, we shall use the indices x; y; z as an alternative to 1; 2; 3, understanding the correspondence x ! 1; y ! 2; z ! 4 Newton’s Laws 23 Tensors. 11) The result is a vector in the direction A, since B C is a scalar. 11) is a linear function of C. A tensor is a quantity whose components transform as a product of the coordinates. For instance, the component Txy of a tensor T transforms as the product xy. The unit tensor is the dyadic I D ii C jj C kk.
In a similar way we can define tensors of third rank as Tijk , etc. For us, the most interesting third rank tensor is the completely antisymmetric unit tensor ijk , called the Levi-Civita tensor. e. , 213; 321; 132). Let us consider two vectors represented by their components Aj and Bk . A2 B3 A3 B2 ; A3 B1 A1 B3 ; A1 B2 A2 B1 /; in other words, Ci with i D 1; 2; 3 are the components of the vector product A B. 15) The pseudovector C is called the dual pseudovector of the tensor T. Very important physical quantities are usually expressed as vector products.