Coordinate and basis changes

Sections in this chapter

Introduction

Coordinate changes

Basis changes

For other chapters go to Index

Introduction

The change of basis mechanism is one of the most interesting tools of TTC and it can save a lot of analytical work. In this chapter we shall explain it, but first of all, we shall give the difference between coordinate change and basis change.

If {e_i } , {eⁱ } and {e'_i },{e'ⁱ } are two pairs of dual basis, then the second one can be expressed in terms of the former as

being n the dimension of the manifold andan matrix . A tensor T can be expressed in each one of the basis. It's components will be related through the matrix . This correspond to the passive point of view . Alternatively we can thing that the basis change is a relation between two manifolds (active point of view). With respect to this point of view there are two tenors T. One for each manifold. This point of view is more adapted at the way you must proceed with TTC and in general with Mathematica. You must discting with to diferents names the tensor T at a diferents basis. Let's name T and T' . The corresponding components are related with

Although the comments with respect the name of tensors above we usually will say that a change of basis has been performed, that is we will adopt the passive point of view

When the change takes place between two coordinate basis and the matrix and its inverse is given by

and often it is said that a coordinate change or coordinate transformation} has been performed.

Note that in a coordinate change, the new components are expressed using the new coordinates.

Coordinate changes

A coordinate change is performed by the operator Change, which can act over scalars, tensors and forms. The relationship between xⁱ and x'ⁱ has to be given by InputCoordinateChange. The jacobian matrix is obtained through JacobianMatrix.

In[2]:= InputCoordinates[cart, {x, y, z}];

In[3]:= InputCoordinates[cyl, {r, phi, z}];

In[4]:= InputCoordinateChange[cart, cyl,
          { x -> r Cos[phi], y -> r Sin[phi], z -> z } ];

Out[4]= {x -> r Cos[phi], y -> r Sin[phi], z -> z}

In[5]:= JacobianMatrix[cart, cyl] // MatrixForm

Out[5]//MatrixForm=  Cos[phi]      -(r Sin[phi])   0

                     Sin[phi]      r Cos[phi]      0

                     0             0               1

Next, we realize the change from cartesian to cylindrical coordinates over a scalar function F=f(x²+y²+z²), over a vector

and over a 1-form

In[6]:= F = f[x^2 + y^2 + z^2]

           2    2    2
Out[6]= f[x  + y  + z ]

In[7]:= Change[cart, cyl][F]

           2    2
Out[7]= f[r  + z ]

In[8]:= e = ZZ[cart];

In[9]:= R = x e[-1] + y e[-2] + z e[-3]

Out[9]= z Dz + y Dy + x Dx

In[10]:= Change[cart, cyl][R]

Out[10]= z Dz + r Dr

In[11]:= w = y e[1] + (x + z Cos[y z]) e[2] + y Cos[y z] e[3]

Out[11]= y dx + (x + z Cos[y z]) dy + y Cos[y z] dz

In[12]:= Change[cart, cyl][w]

Out[12]= (z Cos[r z Sin[phi]] Sin[phi] + r Sin[2 phi]) dr +

>    r (r Cos[2 phi] + z Cos[phi] Cos[r z Sin[phi]]) dphi +

>    r Cos[r z Sin[phi]] Sin[phi] dz

The expression of the euclidean metric in cylindrical coordinates can be obtained easily performing a coordinate change on the metric tensor in cartesian coordinates.

In[13]:= InputMetric[euclidean, cart, e[1, 1] + e[2, 2] + e[3, 3]]

Out[13]= dx*.dx + dy*.dy + dz*.dz

In[14]:= g = %;

In[15]:= Change[cart, cyl][g]

                   2
Out[15]= dr*.dr + r  dphi*.dphi + dz*.dz

In[16]:= InputMetric[euclidean, cyl, %]

                   2
Out[16]= dr*.dr + r  dphi*.dphi + dz*.dz

In[17]:= Metric[euclidean, cyl]

                   2
Out[17]= dr*.dr + r  dphi*.dphi + dz*.dz

The coordinate changes only work in one direction. If you have input the coordinate transformation

with

InputCoordinateChange[x, x', rules]

you cannot change an object expressed in terms of xⁱ to coordinates x'ⁱ\. If you want to do it, the declaration

InputCoordinateChange[x', x, rules]

has to be done. On the contrary, as it will be seen in the next section, the changes of basis allow the transformations to work in both senses.

Basis changes

The basis changes are also performed with \verb+Change+. Nevertheless, in this case, the relation between {e_i} and {e'_i } is entered with InputBasisChange. This command is very flexible and permit to give the relationship between both basis using vectors or forms and in any sense. For instance, in the next example, we give the 1-forms of B1 in terms of those of B2, but we could have given the vectors of B2 in terms of those of B1 and so on. The functions BasisVectorChange and BasisFormChange shows the

relation between both basis. If you are going to perform only algebraic manipulations and the coordinate system involved is of minor importance, the default coordinates XX can be used.

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

In[3]:= InputBasis[{XX, B1}]

Out[3]= {{XX, B1}, {1, 2}}

In[4]:= InputBasis[{XX, B2}]

Out[4]= {{XX, B1}, {1, 2}}
        {{XX, B2}, {1, 2}}

In[5]:= e = ZZ[{XX, B1}]; u = ZZ[{XX, B2}];

In[6]:= InputBasisChange[{XX, B1}, {XX, B2},
             { e[1] -> u[2], e[2] -> 4 u[1] + 8 u[2] } ]

In[7]:= BasisVectorChange[{XX, B1}, {XX, B2}] // ColumnForm

Out[7]= B1[-1] -> B2[-2] - 2 B2[-1]
                  1
        B1[-2] -> - B2[-1]
                  4
In[8]:= BasisFormChange[{XX, B1}, {XX, B2}] // ColumnForm

Out[8]= B1[1] -> B2[2]
        B1[2] -> 4 B2[1] + 8 B2[2]

In[12]:= BasisVectorChange[{XX, B2}, {XX, B1}] // ColumnForm

Out[9]= B2[-1] -> 4 B1[-2]
        B2[-2] -> 8 B1[-2] + B1[-1]

In[10]:= BasisFormChange[{XX, B2}, {XX, B1}] // ColumnForm

                             1
Out[10]= B2[1] -> -2 B1[1] + - B1[2]
                             4
         B2[2] -> B1[1]

In[11]:= t1 = (X1 + X2) e[1, -1] - 3 X1 e[2, -1]

Out[11]= (X1 + X2) B1[1]*.B1[-1] - 3 X1 B1[2]*.B1[-1]

In[12]:= t2 = Change[{XX, B1}, {XX, B2}][t1]

Out[12]= -12 X1 B2[1]*.B2[-2] + 24 X1 B2[1]*.B2[-1] +

>    (-23 X1 + X2) B2[2]*.B2[-2] + (46 X1 - 2 X2) B2[2]*.B2[-1]

In[13]:= Change[{XX, B2}, {XX, B1}][t2]

Out[13]= (-184 X1 + 4 (46 X1 - 2 X2) + 8 X2) B1[1]*.B1[-2] +

>    (X1 + X2) B1[1]*.B1[-1] - 3 X1 B1[2]*.B1[-1]

In[14]:= TensorSimplify[{Expand}][%]

Out[14]= (X1 + X2) B1[1]*.B1[-1] - 3 X1 B1[2]*.B1[-1]

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