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