Quick Introduction to Tensor Analysis

File : pdf, 451 KB, 47 pages

by R. Sharipov

TOC

CONTENTS.

CHAPT I. PRELIMINARY INFORMATION

1. Geometrical and physical vectors

2. Bound vectors and free vectors

3. Euclidean space

4. Bases and Cartesian coordinates

5. What if we need to change a basis ?

6. What happens to vectors when we change the basis ?

7. What is the novelty about vectors that we learned knowing transformation formula for their

coordinates ?

CHAPT II. TENSORS IN CARTESIAN COORDINATES

8. Covectors

9. Scalar product of vector and covector

10. Linear operators

11. Bilinear and quadratic forms

12. General defnition of tensors

13. Dot product and metric tensor

14. Multiplication by numbers and addition

15. Tensor product

16. Contraction

17. Raising and lowering indices

18. Some special tensors and some useful formulas

CHAPT III. TENSOR FIELDS. DIFFERENTIATION OF TENSORS

19. Tensor fields in Cartesian coordinates

20. Change of Cartesian coordinate system

21. Differentiation of tensor fields

22. Gradient, divergency, and rotor Laplace and d’Alambert operators

CHAPT IV. TENSOR FIELDS IN CURVILINEAR COORDINATES

23. General idea of curvilinear coordinates

24. Auxiliary Cartesian coordinate system

25. Coordinate lines and the coordinate grid

26. Moving frame of curvilinear coordinates

27. Dynamics of moving frame

28. Formula for Christophel symbols

29. Tensor fields in curvilinear coordinates

30. Differentiation of tensor fields in curvilinear coordinates

31. Concordance of metric and connection

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