§ 3    Affine Coordinate System

1. Affine coordinate system and metric coefficients

[ Affine coordinates ]   In the three-dimensional Euclidean space V , if a rectangular coordinate system is taken, and its coordinate unit vectors are i , j , and k , the vector a in the space can be expressed as

a = a x i + a y   j + a z k

Generally, given three non-coplanar vectors e 1 , e 2 , e 3 in the space, any vector a in the space can be decomposed according to these three vectors, and let its coefficients be a 1 , a 2 , a 3 ( here 1 , 2, 3 are not exponents, but superscripts ) , then a can be expressed as

a = a 1 e 1 + a 2 e 2 + a 3 e 3

Or simply count as V V a = a i e i

a = a 1 , a 2 , a 3 ={ a i } V V V  Such coordinate systems e 1 , e 2 , e 3 are called affine coordinate systems, e 1 , e 2 , e 3 are called coordinate vectors, and a 1 , a 2 , a 3 are called affine coordinates of vector a .  [ Metric coefficient in Euclidean space ]   When the vector a is written in the above form, its length a is given by

( a ) 2 = ( a i e i )( a j e j ) = ( e i e j ) a i a j

give . order

e i e j = g ij (= g ji )   ( i , j =1,2,3)

Then gij is called the metric coefficient of the affine coordinate system .

1 The length of the vector a is given by ( a ) 2 = g ij a i a j

Calculate .

2 two vectors

a = a i e i , b = b j e j

The included angle is given by cos = Calculate .

3 Since g ij a i a j is a positive definite quadratic form, the determinant made by g ij mixed product

( e 1 , e 2 , e 3 ) 2 = = g ( e 1 , e 2 , e 3 )= [ Kronecker notation ]   Symmetric matrix The inverse matrix of to represent . By the properties of the inverse matrix, there are g ij = g ji and

g ik g kj = in the formula = called Kronecker notation .

[ reciprocal vector ]   use this g ij provision

e i = g ij e j

Hence there is

e j = g ij e i

e i e k =( g ij e j ) e k = g ij ( e j e k )= g ij g jk = e i e j =( g il e l )( g jm e m )= g il g jm ( e l e m )= g il g jm g lm = g il = g ij For e 1 , e 2 , e 3 , we can get

e 1 =( e 2 × e 3 ),

e 2 =( e 3 × e 1 ), e 3 =( e 1 × e 2 )

e 1 , e 2 , e 3 are called reciprocal vectorsabout the coordinate vectors e 1 , e 2 , e 3. g ij is called the metric coefficient inthe affine coordinate system of the reciprocal vectors.

Two, contravariant vector and covariant vector

[ Contravariant vector and covariant vector ]   If the affine coordinates a 1 , a 2 , a 3 of the vector a in the coordinate systems e 1 , e 2 , e 3 are determined by the formula  a = a 1 e 1 + a 2 e 2 + a 3 e 3 = a i e i

Given, a 1 , a 2 , a 3 are called contravariant coordinates ( or called anti-variation coordinates ) of vector a , and vector a i is called a contravariant vector ( or called anti-variation vector ).  If the reciprocal vectors about the coordinate vectors e 1 , e 2 , e 3 are e 1 , e 2 , e 3 , the affine coordinates a 1 , a 2 of the vector a in the coordinate systems e 1 , e 2 , e 3 , a3 is determined by the formula  a = a 1 e 1 + a 2 e 2 + a 3 e 3 = a j e j

given, then a 1 , a 2 , a 3 are called covariant coordinates of vector a, and vector a j is called covariant vector .  In the Cartesian coordinate system, the covariant coordinates and contravariant coordinates of the vector are consistent . Generally, in the affine coordinate system, the covariant coordinates and the contravariant coordinates have a relationship

a i = a · e i =( a j e j ) · e i = a j ( e j · e i )= a j g ji

[ scalar product of contravariant vector and covariant vector ]

If a , b are two vectors, a 1 , a 2 , a 3 ; b 1 , b 2 , b 3 are their contravariant coordinates, respectively, then

a · b = g ij a i b j

If a , b are two vectors, a 1 , a 2 , a 3 ; b 1 , b 2 , b 3 are their covariant coordinates, respectively, then

a · b = g ij a i a j

If the contravariant coordinates of a are a 1 , a 2 , a 3 , and the covariant coordinates of b are b 1 , b 2 , b 3 , then

a · b = a i b i

Three, n -dimensional space

[ Definition of n -dimensional space ]   If a point in the space has a one-to-one correspondence with the values ​​of an ordered group of n independent real numbers x 1 , ···, x n , then, take such a point as an element The set of is called n -dimensional real number space V ( referred to as n -dimensional space ) , denoted as R n . So a point M in the space corresponds to a set of ordered numbers x 1 ,..., x n ; on the contrary, a set of ordered numbers x 1 , ···, x n corresponds to a point M. Such a set of ordered numbers ( x 1 , ···, x n ) is called an n -dimensional spaceThe coordinates of a point M in R n .

[ Vector in n -dimensional space ]   Take a certain point O in the n -dimensional space R n with coordinates (0,0, ··· ,0) , and another point M ( x 1 , x 2 , ···, x n ) , r is a vector corresponding to two points O and M , called the vector radius of point M.

It is assumed that an affine coordinate system can be introduced in R n such that the relationship between the vector radius r and the coordinates of the point M ( x i ) is

r = x 1 e 1 +... + x n e n = x i e i

where e 1 , ···, e n are n linearly independent vectors in R n , and this coordinate system e 1 , ··· , e n is called an affine coordinate system in R n , x 1 , · , x n is called the affine coordinate of the vector r in R n .  Many of the results discussed in the three-dimensional space are valid in the n -dimensional space, as long as the indicators appearing in the formula are considered to be from 1 to n .

[ Contravariant vector and covariant vector ]   Consider an arbitrary coordinate transformation in the n -dimensional space R n V V (1)  where the function has successive derivatives with respect to x i ( the order required in the discussion ) , and the Jacobian of the transformation is not equal to zero:  Therefore (1) has an inverse transform Let a 1 , ··· , a n be the function of x i , if under the coordinate transformation, they are all transformed according to the coordinate differential, that is Then a i is called the contravariant coordinate of a vector in the coordinate system ( x i ) , and it is the contravariant coordinate of the same vector in the coordinate system . The vector is called the contravariant vector .   If a i press form transformation, then a i is called the covariant coordinate of a vector in the coordinate system ( x i ) , and it is called the covariant coordinate of the same vector in the coordinate system, and the vector is called the covariant vector .   The transformation coefficients of contravariant and covariant vectors are different, but there is a relation between them where is the Kronecker notation . The gradient of a scalar field is a covariant vector .

Let the scalar field in n -dimensional space be , its change along an infinitesimal displacement d x i  is an invariant under the coordinate transformation, where is the component of the gradient . Therefore, under the coordinate transformation,   but So it is a covariant vector . V Euclidean space is abbreviated as Euclidean space, and its definition can be found in Chapter 21, §4.

The abbreviation V V is the way it is written in tensor arithmetic.If each indicator appears once in the product, it means it takes all possible values; if

Each indicator appears twice in the product, which means that all possible values ​​are taken, and then the items are added together to find the sum . This rule is called

Agreement for Einstein .

V V V This is the tensor notation.

For another definition of V n -dimensional real number space, see Chapter 21,§3.

V V is used here torepresent the coordinates of the same point M ( xi ) in another coordinate system, that is to say,andsamepoint.

A kernel character ( such as x ) represents the same object, and a prime is added to the index to represent different coordinate systems (such as etc.), this notation is called kernel

standard method .