Coordinate systems and natural basis

Sections in this chapter:

Introduction

Coordinates

Natural basis

For other chapters go to Index

Introduction

For practical purposes, the main subject of a differentiable manifold is the coordinate system that covers it locally. On the other hand, the tensorial objects defined in TTC are expressed using a basis related to a coordinate system, and you have to declare it before doing any other thing.
This chapter describes how to define coordinate systems and introduces the notation to express vectors and 1-forms using natural basis.

Coordinates

TTC there exist a default 3-dimensional coordinate system called XX with coordinate symbols {X1, X2, X3} that do not need tobe declared. In this system, the name and the symbols are fixed and cannot be changed, but its dimension can be established at will using InputCoordinates. To see the symbols associated to a particular coordinate system you can use Coordinates.

In[1]:= Coordinates[XX]

Out[1]= {X1, X2, X3}

In[2]:= InputCoordinates[XX, 4]

Out[2]= {XX, {X1, X2, X3, X4}}

In[3]:= Coordinates[XX]

Out[3]= {X1, X2, X3, X4}

Besides the default coordinate system we can define as much coordinate systems as we want using InputCoordinates. For instance, we can define the cartesian coordinates in a 2-dimensional space with name cart2 and symbols {x, y} and the spherical coordinates sph in a 3-dimensional space with coordinate symbols {r,theta,phi}. A coordinate symbol cannot be used twice in the same system but there is no problem in using an identical symbol in another coordinate system. For example, the system cart2 and the system cart3 with symbols {x, y, z} can be used without problems during the same session.

In[4]:= InputCoordinates[cart2, {x, y}]

Out[4]= {XX, {X1, X2, X3, X4}}
        {cart2, {x, y}}

In[5]:= InputCoordinates[cart3, {x, y, z}]

Out[5]= {XX, {X1, X2, X3, X4}}
        {cart2, {x, y}}
        {cart3, {x, y, z}}

In[6]:= InputCoordinates[sph, {r, theta, phi}]

Out[6]= {XX, {X1, X2, X3, X4}}
        {cart2, {x, y}}
        {cart3, {x, y, z}}
        {sph, {r, theta, phi}}

The output of InputCoordinates shows in column form the coordinate systems defined until now. If we decide to change the symbols of the previous coordinates, two things have to be done:

Clear the old ones using ClearCoordinates

enter them again.

In[7]:= ClearCoordinates[sph]

Out[7]= {XX, {X1, X2, X3, X4}}
        {cart2, {x, y}}
        {cart3, {x, y, z}}

In[8]:= InputCoordinates[sph, {r, th, ph}]

Out[8]= {XX, {X1, X2, X3, X4}}
        {cart2, {x, y}}
        {cart3, {x, y, z}}
        {sph, {r, th, ph}}

Besides clearing the coordinate system, ClearCoordinates has other effects as will be see in How to clear. If we want to know which coordinate systems have been declared so far, we can use CoordinateNames. On the other hand, Dimension yields the dimension of the coordinate system.

In[9]:= CoordinateNames

Out[9]= {XX, {X1, X2, X3, X4}}
        {cart2, {x, y}}
        {cart3, {x, y, z}}
        {sph, {r, th, ph}}

In[10]:= Dimension[sph]

Out[10]= 3

Natural basis

Owing to the way of representing tensors in TTC, one has to give, not only the components, but also the basis elements of every vector, tensor or form. Therefore, a mechanism for expressing the elements of the basis have to be given. In this section we introduce the notation for natural, coordinate or holonom basis and defer the study of noncoordinate basis until Noncoordinate basis .
TTC make use of the name of the coordinates to represent the basis elements of both, vectors and forms. Given a coordinate system, lets say coords, with symbols xⁱ, the name of the natural basis is the same that that of the coordinate system.

IMPORTANT: the symbol used to name the natural basis is the same as that of the corresponding coordinate system.

In consequence, the notation for the basis elements is:

coords[-i]

coords[i]dxⁱ

Instead of i , it is also possible to use xⁱ:

coords[-xⁱ]

coords[xⁱ]dxⁱ

The minus sign is associated with the lower position of the index in . This is a general rule of TTC: a minus sign appears every time you want to indicate that an index is a subscript of a tensor or of a tensor component.

GENERAL RULE: a lower index in both, a basis element or a component of a tensor, is indicated in TTC by a minus sign.

In[1]:= InputCoordinates[cart, {x, y, z}]

Out[1]= {XX, {X1, X2, X3}}
        {cart, {x, y, z}}

In[2]:= XX[-1]

Out[2]= DX1

In[3]:= XX[3]

Out[3]= dX3

In[4]:= cart[-z]

Out[4]= Dz

In[5]:= cart[3]

Out[5]= dz

As you can see, the TTC output imitates the mathematical notation making the substitutions

D and d

d. Next, we define the vector

and the form

In[6]:= cart[x]

Out[6]= dx

In[7]:= v = a x cart[-1] + 3 cart[-y] - y cart[-3]

Out[7]= -y Dz + 3 Dy + a x Dx

In[8]:= w = 5 f[x] cart[2] - cart[z]

Out[8]= 5 f[x] dy - dz

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