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1, This, the Riemann curvature tensor, quantifies space-time curvature.

2, The Ricci tensor has a non-zero divergence which can be removed by a simple subtraction.

3, The dominant term in the stress-energy tensor is therefore the energy density Too.

4, The resulting tensor equation is automatically valid under all general transformations.

5, Under a general transformation which demonstrates that this tensor is the same in all frames.

6, Firstly, physical laws must be expressible as tensor equations so that they remain valid under transformations to any accelerated frame.

7, There are a number of other tensor properties which are needed in Chapters 6 and 7.

8, B is termed the left Cauchy-Green deformation tensor, both terms having obvious origins in the definitions given above.

9, Some differential geometry and tensor analysis is needed in order to appreciate the full flavour of the theory.

10, The curvature tensor can be identified by using an approach that was introduced in Chapter 3.

11, In this case the rotation tensor R is equal to the direction, cosine L. Example 10.

12, The metric tensor requires one transformation for each of its indices.

13, Einstein also recognized that the stress-energy tensor provided the appropriate tensor description for the distribution and flow of energy in space-time.

14, Gravity gradient tensor; Forward; Edge extraction.

15, Conservation of Mass. Conservation of Momentum. Stress Tensor.

16, We derive the relations between physical and tensor components.

17, Tensor algebra is tied to coordinates.

18, But tensor product wavelets has a number of drawbacks.

19, In addition to the usual material[Sentencedict.com],(http://sentencedict.com/tensor.html) it contains useful chapters on electromagnetism and relativistic fluids and introduces the reader to tensor notation.

20, Consequently we can use the quotient theorem to prove that is a tensor.

21, We shall now briefly bring out the relationship between the Gaussian curvature and the Riemann curvature tensor.

22, A more compact way of writing this is where the components form a square matrix: is called a metric tensor.

23, For vacuum solutions, in these regions at most one component, either 4 or 0, of the curvature tensor is non-zero.

24, This leads to the definition of stress as a tensor quantity of rank 2.

25, Appendix B contains more details on the mathematical properties of the Riemann tensor.

26, In Chapter 5 we discovered that physical laws expressed as tensor equations automatically retain their form under general transformations.

27, Applying the usual procedure to obtain this in another frame we obtain so that transforms as a rank 2 tensor.

28, This equation has a different, but still diagonal, metric tensor.

29, A more complete and compact description of curvature in n dimensions is embodied in the Riemann tensor.

30, This can not be a curvature singularity, since the curvature tensor on it is zero.