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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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