**§ ****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**

*a*** = ***a _{x
}i* +

** **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** =

Or simply count as V V ** a** =

** a** =

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* ) ^{2} = ( *a ^{i }e _{i}* )(

give . order

*e ** _{i }e _{j}* =

*Then gij _{}* is called the metric coefficient of the

1 The length of the vector ** a** is given by

( *a* ) ^{2} = *g _{ij} a ^{i} a ^{j}*

Calculate .

2 ^{} two vectors^{}

** a** =

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

_{}

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 ^{ik} g _{kj}* =

in the formula

_{}=_{}

called Kronecker notation .

[ reciprocal vector ] use this *g ^{ij}* provision

** e ^{i}** =

Hence there is_{}

** e _{j}** =

*e ^{i} e *

** e ^{i} e ^{j}** =(

** **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

** a** =

Given, *a *^{1} , *a *^{2} , *a *^{3} are called contravariant coordinates ( or called anti-variation coordinates ) of vector ** a** , and 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

** a** =

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}* =

[ scalar product of contravariant vector and covariant vector ]

** **If ** a** ,

*a*** · b** =

If ** a** ,

*a*** · b** =

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** =

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

[ 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

It is assumed that an affine coordinate system can be introduced in *R ^{n such that the relationship between the vector radius }*

** r** =

where *e*_{ 1} , ···, *e ** _{n}* are

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

_{}

*Then a ^{i}* is called the contravariant coordinate of a vector in the coordinate system (

If *a _{i}* press

_{}

form transformation, then *a ^{i}* is called the covariant coordinate of a vector in the coordinate system (

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 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 .

Contribute a better translation