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Ⅰ．FORMAL PRELIMINARIES 1

1．The summation convention 1

2，3．The Kronecker deltas 3

4．Linear equations 6

5，6．Functional Determinants 7

7．Derivative of a determinant 8

8．Numerical relations 8

9，10．Minors，cofactors，and the Laplace expansion 9

11．Historical 11

Ⅱ．DIFFERENTIAL INVARIANTS 13

1．N-dimensional space 13

2．Transformations of coordinates 13

3．Invariants 14

4．Differential invariants 15

5，6．Differentials and contravariant vectors 16

7．A general class of invariants 19

8．Tensors 19

9．Relative scalars 20

10．Covariant vectors 21

11，12．Algebraic combinations of tensors 22

13．The commonness of tensors 24

14．Numerical tensors 25

15．Combinations of vectors 26

16．Historical and general remarks 27

Ⅲ．QUADRATIC DIFFERENTIAL FORMS 30

1．Differential forms 30

2．Linear differential forms 30

3，4．Quadratic differential forms 31

5．Invariants derived from basic invariants 32

6，7．Invariants of a quadratic differential form 32

8，9．The fundamental affine connection 33

10．Affine connections in general 35

11，12．Covariant differentiation 36

13．Geodesic coordinates 38

14，15．Formulas of covariant differentiation 39

16，17．The curvature tensor 41

18，19．Riemann-Christoffel tensor 43

20．Reduction theorems 44

21．Historical remarks 47

22．Scalar invariants 48

Ⅳ．EUCLIDEAN GEOMETRY 50

1，2．Euclidean geometry 50

3．Euclidean affine geometry 53

4，5．Euclidean vector analysis 55

6．Associated vectors and tensors 56

7．Distance and scalar product 57

8，9．Area 59

10．First order differential parameters 60

11．Euclidean covariant differentiation 61

12．The divergence 62

13．The Laplacian or Lamé differential parameter of the second order 63

14．The curl of a vector 64

15，16．Generalized divergence and curl 64

17．Historical remarks 66

Ⅴ．THE EQUIVALENCE PROBLEM 67

1．Riemannian geometry 67

2．The theory of surfaces 68

3．Spaces immersed in a Euclidean space 69

4，5．Condition that a Riemannian space be Euclidean 69

6．The equivalence problem 72

7．A lemma on mixed systems 73

8．Equivalence theorem for quadratic differential forms 76

9．Equivalence of affine connections 77

10，11．Automorphisms of a quadratic differential form 78

12．Equivalence theorem in terms of scalars 79

13．Historical remarks 80

Ⅵ．NORMAL COORDINATES 82

1，2．Affine geometry of paths 82

3，4．Affine normal coordinates 85

5．Affine extensions 87

6．The affine normal tensors 89

7，8．The replacement theorems 90

9，10，11，12．The curvature tensor and the normal tensors 91

13，14，15，16，17．Affine extensions of the fundamental tensor 94

18．Historical and general remarks 100

19．Formulas for the extensions of tensors 102