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# 28. itensor

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## 28.1 Introduction to itensor

Maxima implements symbolic tensor manipulation of two distinct types: component tensor manipulation (`ctensor` package) and indicial tensor manipulation (`itensor` package).

Nota bene: Please see the note on 'new tensor notation' below.

Component tensor manipulation means that geometrical tensor objects are represented as arrays or matrices. Tensor operations such as contraction or covariant differentiation are carried out by actually summing over repeated (dummy) indices with `do` statements. That is, one explicitly performs operations on the appropriate tensor components stored in an array or matrix.

Indicial tensor manipulation is implemented by representing tensors as functions of their covariant, contravariant and derivative indices. Tensor operations such as contraction or covariant differentiation are performed by manipulating the indices themselves rather than the components to which they correspond.

These two approaches to the treatment of differential, algebraic and analytic processes in the context of Riemannian geometry have various advantages and disadvantages which reveal themselves only through the particular nature and difficulty of the user's problem. However, one should keep in mind the following characteristics of the two implementations:

The representation of tensors and tensor operations explicitly in terms of their components makes `ctensor` easy to use. Specification of the metric and the computation of the induced tensors and invariants is straightforward. Although all of Maxima's powerful simplification capacity is at hand, a complex metric with intricate functional and coordinate dependencies can easily lead to expressions whose size is excessive and whose structure is hidden. In addition, many calculations involve intermediate expressions which swell causing programs to terminate before completion. Through experience, a user can avoid many of these difficulties.

Because of the special way in which tensors and tensor operations are represented in terms of symbolic operations on their indices, expressions which in the component representation would be unmanageable can sometimes be greatly simplified by using the special routines for symmetrical objects in `itensor`. In this way the structure of a large expression may be more transparent. On the other hand, because of the the special indicial representation in `itensor`, in some cases the user may find difficulty with the specification of the metric, function definition, and the evaluation of differentiated "indexed" objects.

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### 28.1.1 New tensor notation

Until now, the `itensor` package in Maxima has used a notation that sometimes led to incorrect index ordering. Consider the following, for instance:

    (%i2) imetric(g);
    (%o2)                                done
    (%i3) ishow(g([],[j,k])*g([],[i,l])*a([i,j],[]))$
                                     i l  j k
    (%t3)                           g    g    a
                                               i j
    (%i4) ishow(contract(%))$
                                          k l
    (%t4)                                a

This result is incorrect unless `a` happens to be a symmetric tensor. The reason why this happens is that although `itensor` correctly maintains the order within the set of covariant and contravariant indices, once an index is raised or lowered, its position relative to the other set of indices is lost.

To avoid this problem, a new notation has been developed that remains fully compatible with the existing notation and can be used interchangeably. In this notation, contravariant indices are inserted in the appropriate positions in the covariant index list, but with a minus sign prepended. Functions like `contract` and `ishow` are now aware of this new index notation and can process tensors appropriately.

In this new notation, the previous example yields a correct result:

    (%i5) ishow(g([-j,-k],[])*g([-i,-l],[])*a([i,j],[]))$
                                     i l       j k
    (%t5)                           g    a    g
                                          i j
    (%i6) ishow(contract(%))$
                                          l k
    (%t6)                                a

Presently, the only code that makes use of this notation is the `lc2kdt` function. Through this notation, it achieves consistent results as it applies the metric tensor to resolve Levi-Civita symbols without resorting to numeric indices.

Since this code is brand new, it probably contains bugs. While it has been tested to make sure that it doesn't break anything using the "old" tensor notation, there is a considerable chance that "new" tensors will fail to interoperate with certain functions or features. These bugs will be fixed as they are encountered... until then, caveat emptor!

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### 28.1.2 Indicial tensor manipulation

The indicial tensor manipulation package may be loaded by `load(itensor)`. Demos are also available: try `demo(tensor)`.

In `itensor` a tensor is represented as an "indexed object" . This is a function of 3 groups of indices which represent the covariant, contravariant and derivative indices. The covariant indices are specified by a list as the first argument to the indexed object, and the contravariant indices by a list as the second argument. If the indexed object lacks either of these groups of indices then the empty list `[]` is given as the corresponding argument. Thus, `g([a,b],[c])` represents an indexed object called `g` which has two covariant indices `(a,b)`, one contravariant index (`c`) and no derivative indices.

The derivative indices, if they are present, are appended as additional arguments to the symbolic function representing the tensor. They can be explicitly specified by the user or be created in the process of differentiation with respect to some coordinate variable. Since ordinary differentiation is commutative, the derivative indices are sorted alphanumerically, unless `iframe_flag` is set to `true`, indicating that a frame metric is being used. This canonical ordering makes it possible for Maxima to recognize that, for example, `t([a],[b],i,j)` is the same as `t([a],[b],j,i)`. Differentiation of an indexed object with respect to some coordinate whose index does not appear as an argument to the indexed object would normally yield zero. This is because Maxima would not know that the tensor represented by the indexed object might depend implicitly on the corresponding coordinate. By modifying the existing Maxima function `diff` in `itensor`, Maxima now assumes that all indexed objects depend on any variable of differentiation unless otherwise stated. This makes it possible for the summation convention to be extended to derivative indices. It should be noted that `itensor` does not possess the capabilities of raising derivative indices, and so they are always treated as covariant.

The following functions are available in the tensor package for manipulating indexed objects. At present, with respect to the simplification routines, it is assumed that indexed objects do not by default possess symmetry properties. This can be overridden by setting the variable `allsym[false]` to `true`, which will result in treating all indexed objects completely symmetric in their lists of covariant indices and symmetric in their lists of contravariant indices.

The `itensor` package generally treats tensors as opaque objects. Tensorial equations are manipulated based on algebraic rules, specifically symmetry and contraction rules. In addition, the `itensor` package understands covariant differentiation, curvature, and torsion. Calculations can be performed relative to a metric of moving frame, depending on the setting of the `iframe_flag` variable.

A sample session below demonstrates how to load the `itensor` package, specify the name of the metric, and perform some simple calculations.

    (%i1) load(itensor);
    (%o1)      /share/tensor/itensor.lisp
    (%i2) imetric(g);
    (%o2)                                done
    (%i3) components(g([i,j],[]),p([i,j],[])*e([],[]))$
    (%i4) ishow(g([k,l],[]))$
    (%t4)                               e p
                                           k l
    (%i5) ishow(diff(v([i],[]),t))$
    (%t5)                                  0
    (%i6) depends(v,t);
    (%o6)                               [v(t)]
    (%i7) ishow(diff(v([i],[]),t))$
                                        d
    (%t7)                               -- (v )
                                        dt   i
    (%i8) ishow(idiff(v([i],[]),j))$
    (%t8)                                v
                                          i,j
    (%i9) ishow(extdiff(v([i],[]),j))$
    (%t9)                             v    - v
                                       j,i    i,j
                                      -----------
                                           2
    (%i10) ishow(liediff(v,w([i],[])))$
                                   %3          %3
    (%t10)                        v   w     + v   w
                                       i,%3    ,i  %3
    (%i11) ishow(covdiff(v([i],[]),j))$
                                                  %4
    (%t11)                        v    - v   ichr2
                                   i,j    %4      i j
    (%i12) ishow(ev(%,ichr2))$
                   %4 %5
    (%t12) v    - g      v   (e p       + e   p     - e p       - e    p
            i,j           %4     j %5,i    ,i  j %5      i j,%5    ,%5  i j

                                                    + e p       + e   p    )/2
                                                         i %5,j    ,j  i %5
    (%i13) iframe_flag:true;
    (%o13)                               true
    (%i14) ishow(covdiff(v([i],[]),j))$
                                                 %6
    (%t14)                        v    - v   icc2
                                   i,j    %6     i j
    (%i15) ishow(ev(%,icc2))$
                                                 %6
    (%t15)                        v    - v   ifc2
                                   i,j    %6     i j
    (%i16) ishow(radcan(ev(%,ifc2,ifc1)))$
                 %6 %8                    %6 %8
    (%t16) - (ifg      v   ifb       + ifg      v   ifb       - 2 v
                        %6    j %8 i             %6    i j %8      i,j

                                                        %6 %8
                                                   - ifg      v   ifb      )/2
                                                               %6    %8 i j
    (%i17) ishow(canform(s([i,j],[])-s([j,i])))$
    (%t17)                            s    - s
                                       i j    j i
    (%i18) decsym(s,2,0,[sym(all)],[]);
    (%o18)                               done
    (%i19) ishow(canform(s([i,j],[])-s([j,i])))$
    (%t19)                                 0
    (%i20) ishow(canform(a([i,j],[])+a([j,i])))$
    (%t20)                            a    + a
                                       j i    i j
    (%i21) decsym(a,2,0,[anti(all)],[]);
    (%o21)                               done
    (%i22) ishow(canform(a([i,j],[])+a([j,i])))$
    (%t22)                                 0

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## 28.2 Definitions for itensor

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### 28.2.1 Managing indexed objects

Function: entertensor (name)

  is a function which, by prompting, allows one to create an indexed object called name with any number of tensorial and derivative indices. Either a single index or a list of indices (which may be null) is acceptable input (see the example under `covdiff`).

Function: changename (old, new, expr)

  will change the name of all indexed objects called old to new in expr. old may be either a symbol or a list of the form `[name, m, n]` in which case only those indexed objects called name with m covariant and n contravariant indices will be renamed to new.

Function: listoftens

  Lists all tensors in a tensorial expression, complete with their indices. E.g.,

      (%i6) ishow(a([i,j],[k])*b([u],[],v)+c([x,y],[])*d([],[])*e)$
                                               k
      (%t6)                        d e c    + a    b
                                        x y    i j  u,v
      (%i7) ishow(listoftens(%))$
                                     k
      (%t7)                        [a   , b   , c   , d]
                                     i j   u,v   x y

Function: ishow (expr)

  displays expr with the indexed objects in it shown having their covariant indices as subscripts and contravariant indices as superscripts. The derivative indices are displayed as subscripts, separated from the covariant indices by a comma (see the examples throughout this document).

Function: indices (expr)

  Returns a list of two elements. The first is a list of the free indices in expr (those that occur only once). The second is the list of the dummy indices in expr (those that occur exactly twice) as the following example demonstrates.

      (%i1) load(itensor);
      (%o1)      /share/tensor/itensor.lisp
      (%i2) ishow(a([i,j],[k,l],m,n)*b([k,o],[j,m,p],q,r))$
                                      k l      j m p
      (%t2)                          a        b
                                      i j,m n  k o,q r
      (%i3) indices(%);
      (%o3)                 [[l, p, i, n, o, q, r], [k, j, m]]

  A tensor product containing the same index more than twice is syntactically illegal. `indices` attempts to deal with these expressions in a reasonable manner; however, when it is called to operate upon such an illegal expression, its behavior should be considered undefined.

Function: rename (expr)
Function: rename (expr, count)

  Returns an expression equivalent to expr but with the dummy indices in each term chosen from the set `[%1, %2,...]`, if the optional second argument is omitted. Otherwise, the dummy indices are indexed beginning at the value of count. Each dummy index in a product will be different. For a sum, `rename` will operate upon each term in the sum resetting the counter with each term. In this way `rename` can serve as a tensorial simplifier. In addition, the indices will be sorted alphanumerically (if `allsym` is `true`) with respect to covariant or contravariant indices depending upon the value of `flipflag`. If `flipflag` is `false` then the indices will be renamed according to the order of the contravariant indices. If `flipflag` is `true` the renaming will occur according to the order of the covariant indices. It often happens that the combined effect of the two renamings will reduce an expression more than either one by itself.

      (%i1) load(itensor);
      (%o1)      /share/tensor/itensor.lisp
      (%i2) allsym:true;
      (%o2)                                true
      (%i3) g([],[%4,%5])*g([],[%6,%7])*ichr2([%1,%4],[%3])*
      ichr2([%2,%3],[u])*ichr2([%5,%6],[%1])*ichr2([%7,r],[%2])-
      g([],[%4,%5])*g([],[%6,%7])*ichr2([%1,%2],[u])*
      ichr2([%3,%5],[%1])*ichr2([%4,%6],[%3])*ichr2([%7,r],[%2]),noeval$
      (%i4) expr:ishow(%)$

             %4 %5  %6 %7      %3         u          %1         %2
      (%t4) g      g      ichr2      ichr2      ichr2      ichr2
                               %1 %4      %2 %3      %5 %6      %7 r

                    %4 %5  %6 %7      u          %1         %3         %2
                 - g      g      ichr2      ichr2      ichr2      ichr2
                                      %1 %2      %3 %5      %4 %6      %7 r
      (%i5) flipflag:true;
      (%o5)                                true
      (%i6) ishow(rename(expr))$
             %2 %5  %6 %7      %4         u          %1         %3
      (%t6) g      g      ichr2      ichr2      ichr2      ichr2
                               %1 %2      %3 %4      %5 %6      %7 r

                    %4 %5  %6 %7      u          %1         %3         %2
                 - g      g      ichr2      ichr2      ichr2      ichr2
                                      %1 %2      %3 %4      %5 %6      %7 r
      (%i7) flipflag:false;
      (%o7)                                false
      (%i8) rename(%th(2));
      (%o8)                                  0
      (%i9) ishow(rename(expr))$
             %1 %2  %3 %4      %5         %6         %7        u
      (%t9) g      g      ichr2      ichr2      ichr2     ichr2
                               %1 %6      %2 %3      %4 r      %5 %7

                    %1 %2  %3 %4      %6         %5         %7        u
                 - g      g      ichr2      ichr2      ichr2     ichr2
                                      %1 %3      %2 %6      %4 r      %5 %7

Option variable: flipflag

  Default: `false`. If `false` then the indices will be renamed according to the order of the contravariant indices, otherwise according to the order of the covariant indices.

  If `flipflag` is `false` then `rename` forms a list of the contravariant indices as they are encountered from left to right (if `true` then of the covariant indices). The first dummy index in the list is renamed to `%1`, the next to `%2`, etc. Then sorting occurs after the `rename`-ing (see the example under `rename`).

Function: defcon (tensor_1)
Function: defcon (tensor_1, tensor_2, tensor_3)

  gives tensor_1 the property that the contraction of a product of tensor_1 and tensor_2 results in tensor_3 with the appropriate indices. If only one argument, tensor_1, is given, then the contraction of the product of tensor_1 with any indexed object having the appropriate indices (say `my_tensor`) will yield an indexed object with that name, i.e. `my_tensor`, and with a new set of indices reflecting the contractions performed. For example, if `imetric:g`, then `defcon(g)` will implement the raising and lowering of indices through contraction with the metric tensor. More than one `defcon` can be given for the same indexed object; the latest one given which applies in a particular contraction will be used. `contractions` is a list of those indexed objects which have been given contraction properties with `defcon`.

Function: remcon (tensor_1, ..., tensor_n)
Function: remcon (all)

  removes all the contraction properties from the tensor_1, ..., tensor_n). `remcon(all)` removes all contraction properties from all indexed objects.

Function: contract (expr)

  Carries out the tensorial contractions in expr which may be any combination of sums and products. This function uses the information given to the `defcon` function. For best results, `expr` should be fully expanded. `ratexpand` is the fastest way to expand products and powers of sums if there are no variables in the denominators of the terms. The `gcd` switch should be `false` if GCD cancellations are unnecessary.

Function: indexed_tensor (tensor)

  Must be executed before assigning components to a tensor for which a built in value already exists as with `ichr1`, `ichr2`, `icurvature`. See the example under `icurvature`.

Function: components (tensor, expr)

  permits one to assign an indicial value to an expression expr giving the values of the components of tensor. These are automatically substituted for the tensor whenever it occurs with all of its indices. The tensor must be of the form `t([...],[...])` where either list may be empty. expr can be any indexed expression involving other objects with the same free indices as tensor. When used to assign values to the metric tensor wherein the components contain dummy indices one must be careful to define these indices to avoid the generation of multiple dummy indices. Removal of this assignment is given to the function `remcomps`.

  It is important to keep in mind that `components` cares only about the valence of a tensor, not about any particular index ordering. Thus assigning components to, say, `x([i,-j],[])`, `x([-j,i],[])`, or `x([i],[j])` all produce the same result, namely components being assigned to a tensor named `x` with valence `(1,1)`.

  Components can be assigned to an indexed expression in four ways, two of which involve the use of the `components` command:

  1) As an indexed expression. For instance:

      (%i2) components(g([],[i,j]),e([],[i])*p([],[j]))$
      (%i3) ishow(g([],[i,j]))$
                                            i  j
      (%t3)                                e  p

  2) As a matrix:

      (%i6) components(g([i,j],[]),lg);
      (%o6)                                done
      (%i7) ishow(g([i,j],[]))$
      (%t7)                                g
                                            i j
      (%i8) g([3,3],[]);
      (%o8)                                  1
      (%i9) g([4,4],[]);
      (%o9)                                 - 1

  3) As a function. You can use a Maxima function to specify the components of a tensor based on its indices. For instance, the following code assigns `kdelta` to `h` if `h` has the same number of covariant and contravariant indices and no derivative indices, and `g` otherwise:

      (%i4) h(l1,l2,[l3]):=if length(l1)=length(l2) and length(l3)=0
        then kdelta(l1,l2) else apply(g,append([l1,l2], l3))$
      (%i5) ishow(h([i],[j]))$
                                                j
      (%t5)                               kdelta
                                                i
      (%i6) ishow(h([i,j],[k],l))$
                                           k
      (%t6)                               g
                                           i j,l

  4) Using Maxima's pattern matching capabilities, specifically the `defrule` and `applyb1` commands:

      (%i1) load(itensor);
      (%o1)      /share/tensor/itensor.lisp
      (%i2) matchdeclare(l1,listp);
      (%o2)                                done
      (%i3) defrule(r1,m(l1,[]),(i1:idummy(),
            g([l1[1],l1[2]],[])*q([i1],[])*e([],[i1])))$

      (%i4) defrule(r2,m([],l1),(i1:idummy(),
            w([],[l1[1],l1[2]])*e([i1],[])*q([],[i1])))$

      (%i5) ishow(m([i,n],[])*m([],[i,m]))$
                                          i m
      (%t5)                              m    m
                                               i n
      (%i6) ishow(rename(applyb1(%,r1,r2)))$
                                 %1  %2  %3 m
      (%t6)                     e   q   w     q   e   g
                                               %1  %2  %3 n

Function: remcomps (tensor)

  Unbinds all values from tensor which were assigned with the `components` function.

Function: showcomps (tensor)

  Shows component assignments of a tensor, as made using the `components` command. This function can be particularly useful when a matrix is assigned to an indicial tensor using `components`, as demonstrated by the following example:

      (%i1) load(ctensor);
      (%o1)       /share/tensor/ctensor.mac
      (%i2) load(itensor);
      (%o2)      /share/tensor/itensor.lisp
      (%i3) lg:matrix([sqrt(r/(r-2*m)),0,0,0],[0,r,0,0],
                      [0,0,sin(theta)*r,0],[0,0,0,sqrt((r-2*m)/r)]);
                     [         r                                     ]
                     [ sqrt(-------)  0       0              0       ]
                     [      r - 2 m                                  ]
                     [                                               ]
                     [       0        r       0              0       ]
      (%o3)          [                                               ]
                     [       0        0  r sin(theta)        0       ]
                     [                                               ]
                     [                                      r - 2 m  ]
                     [       0        0       0        sqrt(-------) ]
                     [                                         r     ]
      (%i4) components(g([i,j],[]),lg);
      (%o4)                                done
      (%i5) showcomps(g([i,j],[]));
                        [         r                                     ]
                        [ sqrt(-------)  0       0              0       ]
                        [      r - 2 m                                  ]
                        [                                               ]
                        [       0        r       0              0       ]
      (%t5)      g    = [                                               ]
                  i j   [       0        0  r sin(theta)        0       ]
                        [                                               ]
                        [                                      r - 2 m  ]
                        [       0        0       0        sqrt(-------) ]
                        [                                         r     ]
      (%o5)                                false

  The `showcomps` command can also display components of a tensor of rank higher than 2.

Function: idummy ()

  Increments `icounter` and returns as its value an index of the form `%n` where n is a positive integer. This guarantees that dummy indices which are needed in forming expressions will not conflict with indices already in use (see the example under `indices`).

Option variable: idummyx

  Default value: `%`

  Is the prefix for dummy indices (see the example under `indices`).

Option variable: icounter

  Default value: `1`

  Determines the numerical suffix to be used in generating the next dummy index in the tensor package. The prefix is determined by the option `idummy` (default: `%`).

Function: kdelta (L1, L2)

  is the generalized Kronecker delta function defined in the `itensor` package with L1 the list of covariant indices and L2 the list of contravariant indices. `kdelta([i],[j])` returns the ordinary Kronecker delta. The command `ev(expr,kdelta)` causes the evaluation of an expression containing `kdelta([],[])` to the dimension of the manifold.

  In what amounts to an abuse of this notation, `itensor` also allows `kdelta` to have 2 covariant and no contravariant, or 2 contravariant and no covariant indices, in effect providing a co(ntra)variant "unit matrix" capability. This is strictly considered a programming aid and not meant to imply that `kdelta([i,j],[])` is a valid tensorial object.

Function: kdels (L1, L2)

  Symmetricized Kronecker delta, used in some calculations. For instance:

      (%i1) load(itensor);
      (%o1)      /share/tensor/itensor.lisp
      (%i2) kdelta([1,2],[2,1]);
      (%o2)                                 - 1
      (%i3) kdels([1,2],[2,1]);
      (%o3)                                  1
      (%i4) ishow(kdelta([a,b],[c,d]))$
                                   c       d         d       c
      (%t4)                  kdelta  kdelta  - kdelta  kdelta
                                   a       b         a       b
      (%i4) ishow(kdels([a,b],[c,d]))$
                                   c       d         d       c
      (%t4)                  kdelta  kdelta  + kdelta  kdelta
                                   a       b         a       b

Function: levi_civita (L)

  is the permutation (or Levi-Civita) tensor which yields 1 if the list L consists of an even permutation of integers, -1 if it consists of an odd permutation, and 0 if some indices in L are repeated.

Function: lc2kdt (expr)

  Simplifies expressions containing the Levi-Civita symbol, converting these to Kronecker-delta expressions when possible. The main difference between this function and simply evaluating the Levi-Civita symbol is that direct evaluation often results in Kronecker expressions containing numerical indices. This is often undesirable as it prevents further simplification. The `lc2kdt` function avoids this problem, yielding expressions that are more easily simplified with `rename` or `contract`.

      (%i1) load(itensor);
      (%o1)      /share/tensor/itensor.lisp
      (%i2) expr:ishow('levi_civita([],[i,j])*'levi_civita([k,l],[])*a([j],[k]))$
                                        i j  k
      (%t2)                  levi_civita    a  levi_civita
                                             j            k l
      (%i3) ishow(ev(expr,levi_civita))$
                                        i j  k       1 2
      (%t3)                       kdelta    a  kdelta
                                        1 2  j       k l
      (%i4) ishow(ev(%,kdelta))$
                   i       j         j       i   k
      (%t4) (kdelta  kdelta  - kdelta  kdelta ) a
                   1       2         1       2   j

                                     1       2         2       1
                              (kdelta  kdelta  - kdelta  kdelta )
                                     k       l         k       l
      (%i5) ishow(lc2kdt(expr))$
                           k       i       j    k       j       i
      (%t5)               a  kdelta  kdelta  - a  kdelta  kdelta
                           j       k       l    j       k       l
      (%i6) ishow(contract(expand(%)))$
                                       i           i
      (%t6)                           a  - a kdelta
                                       l           l

  The `lc2kdt` function sometimes makes use of the metric tensor. If the metric tensor was not defined previously with `imetric`, this results in an error.

      (%i7) expr:ishow('levi_civita([],[i,j])*'levi_civita([],[k,l])*a([j,k],[]))$
                                       i j            k l
      (%t7)                 levi_civita    levi_civita    a
                                                           j k
      (%i8) ishow(lc2kdt(expr))$
      Maxima encountered a Lisp error:

       Error in $IMETRIC [or a callee]:
       $IMETRIC [or a callee] requires less than two arguments.

      Automatically continuing.
      To reenable the Lisp debugger set *debugger-hook* to nil.
      (%i9) imetric(g);
      (%o9)                                done
      (%i10) ishow(lc2kdt(expr))$
               %3 i       k   %4 j       l     %3 i       l   %4 j       k
      (%t10) (g     kdelta   g     kdelta   - g     kdelta   g     kdelta  ) a
                          %3             %4               %3             %4   j k
      (%i11) ishow(contract(expand(%)))$
                                        l i      l i
      (%t11)                           a    - a g

Function: lc_l

  Simplification rule used for expressions containing the unevaluated Levi-Civita symbol (`levi_civita`). Along with `lc_u`, it can be used to simplify many expressions more efficiently than the evaluation of `levi_civita`. For example:

      (%i1) load(itensor);
      (%o1)      /share/tensor/itensor.lisp
      (%i2)  el1:ishow('levi_civita([i,j,k],[])*a([],[i])*a([],[j]))$
                                   i  j
      (%t2)                       a  a  levi_civita
                                                   i j k
      (%i3) el2:ishow('levi_civita([],[i,j,k])*a([i])*a([j]))$
                                             i j k
      (%t3)                       levi_civita      a  a
                                                    i  j
      (%i4) ishow(canform(contract(expand(applyb1(el1,lc_l,lc_u)))))$
      (%t4)                                  0
      (%i5) ishow(canform(contract(expand(applyb1(el2,lc_l,lc_u)))))$
      (%t5)                                  0

Function: lc_u

  Simplification rule used for expressions containing the unevaluated Levi-Civita symbol (`levi_civita`). Along with `lc_u`, it can be used to simplify many expressions more efficiently than the evaluation of `levi_civita`. For details, see `lc_l`.

Function: canten (expr)

  Simplifies expr by renaming (see `rename`) and permuting dummy indices. `rename` is restricted to sums of tensor products in which no derivatives are present. As such it is limited and should only be used if `canform` is not capable of carrying out the required simplification.

  The `canten` function returns a mathematically correct result only if its argument is an expression that is fully symmetric in its indices. For this reason, `canten` returns an error if `allsym` is not set to `true`.

Function: concan (expr)

  Similar to `canten` but also performs index contraction.

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### 28.2.2 Tensor symmetries

Option variable: allsym

  Default: `false`. if `true` then all indexed objects are assumed symmetric in all of their covariant and contravariant indices. If `false` then no symmetries of any kind are assumed in these indices. Derivative indices are always taken to be symmetric unless `iframe_flag` is set to `true`.

Function: decsym (tensor, m, n, [cov_1, cov_2, ...], [contr_1, contr_2, ...])

  Declares symmetry properties for tensor of m covariant and n contravariant indices. The cov_i and contr_i are pseudofunctions expressing symmetry relations among the covariant and contravariant indices respectively. These are of the form `symoper(index_1, index_2,...)` where `symoper` is one of `sym`, `anti` or `cyc` and the index_i are integers indicating the position of the index in the tensor. This will declare tensor to be symmetric, antisymmetric or cyclic respectively in the index_i. `symoper(all)` is also an allowable form which indicates all indices obey the symmetry condition. For example, given an object `b` with 5 covariant indices, `decsym(b,5,3,[sym(1,2),anti(3,4)],[cyc(all)])` declares `b` symmetric in its first and second and antisymmetric in its third and fourth covariant indices, and cyclic in all of its contravariant indices. Either list of symmetry declarations may be null. The function which performs the simplifications is `canform` as the example below illustrates.

      (%i1) load(itensor);
      (%o1)      /share/tensor/itensor.lisp
      (%i2) expr:contract(expand(a([i1,j1,k1],[])*kdels([i,j,k],[i1,j1,k1])))$
      (%i3) ishow(expr)$
      (%t3)         a      + a      + a      + a      + a      + a
                     k j i    k i j    j k i    j i k    i k j    i j k
      (%i4) decsym(a,3,0,[sym(all)],[]);
      (%o4)                                done
      (%i5) ishow(canform(expr))$
      (%t5)                              6 a
                                            i j k
      (%i6) remsym(a,3,0);
      (%o6)                                done
      (%i7) decsym(a,3,0,[anti(all)],[]);
      (%o7)                                done
      (%i8) ishow(canform(expr))$
      (%t8)                                  0
      (%i9) remsym(a,3,0);
      (%o9)                                done
      (%i10) decsym(a,3,0,[cyc(all)],[]);
      (%o10)                               done
      (%i11) ishow(canform(expr))$
      (%t11)                        3 a      + 3 a
                                       i k j      i j k
      (%i12) dispsym(a,3,0);
      (%o12)                     [[cyc, [[1, 2, 3]], []]]

Function: remsym (tensor, m, n)

  Removes all symmetry properties from tensor which has m covariant indices and n contravariant indices.

Function: canform (expr)

  Simplifies expr by renaming dummy indices and reordering all indices as dictated by symmetry conditions imposed on them. If `allsym` is `true` then all indices are assumed symmetric, otherwise symmetry information provided by `decsym` declarations will be used. The dummy indices are renamed in the same manner as in the `rename` function. When `canform` is applied to a large expression the calculation may take a considerable amount of time. This time can be shortened by calling `rename` on the expression first. Also see the example under `decsym`. Note: `canform` may not be able to reduce an expression completely to its simplest form although it will always return a mathematically correct result.

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### 28.2.3 Indicial tensor calculus

Function: diff (expr, v_1, [n_1, [v_2, n_2] ...])

  is the usual Maxima differentiation function which has been expanded in its abilities for `itensor`. It takes the derivative of expr with respect to v_1 n_1 times, with respect to v_2 n_2 times, etc. For the tensor package, the function has been modified so that the v_i may be integers from 1 up to the value of the variable `dim`. This will cause the differentiation to be carried out with respect to the v_ith member of the list `vect_coords`. If `vect_coords` is bound to an atomic variable, then that variable subscripted by v_i will be used for the variable of differentiation. This permits an array of coordinate names or subscripted names like `x[1]`, `x[2]`, ... to be used.

Function: idiff (expr, v_1, [n_1, [v_2, n_2] ...])

  Indicial differentiation. Unlike `diff`, which differentiates with respect to an independent variable, `idiff)` can be used to differentiate with respect to a coordinate. For an indexed object, this amounts to appending the v_i as derivative indices. Subsequently, derivative indices will be sorted, unless `iframe_flag` is set to `true`.

  `idiff` can also differentiate the determinant of the metric tensor. Thus, if `imetric` has been bound to `G` then `idiff(determinant(g),k)` will return `2*determinant(g)*ichr2([%i,k],[%i])` where the dummy index `%i` is chosen appropriately.

Function: liediff (v, ten)

  Computes the Lie-derivative of the tensorial expression ten with respect to the vector field v. ten should be any indexed tensor expression; v should be the name (without indices) of a vector field. For example:

      (%i1) load(itensor);
      (%o1)      /share/tensor/itensor.lisp
      (%i2) ishow(liediff(v,a([i,j],[])*b([],[k],l)))$
             k    %2            %2          %2
      (%t2) b   (v   a       + v   a     + v   a    )
             ,l       i j,%2    ,j  i %2    ,i  %2 j

                                      %1  k        %1  k      %1  k
                                  + (v   b      - b   v    + v   b   ) a
                                          ,%1 l    ,l  ,%1    ,l  ,%1   i j

Function: rediff (ten)

  Evaluates all occurrences of the `idiff` command in the tensorial expression ten.

Function: undiff (expr)

  Returns an expression equivalent to expr but with all derivatives of indexed objects replaced by the noun form of the `idiff` function. Its arguments would yield that indexed object if the differentiation were carried out. This is useful when it is desired to replace a differentiated indexed object with some function definition resulting in expr and then carry out the differentiation by saying `ev(expr, idiff)`.

Function: evundiff (expr)

  Equivalent to the execution of `undiff`, followed by `ev` and `rediff`.

  The point of this operation is to easily evalute expressions that cannot be directly evaluated in derivative form. For instance, the following causes an error:

      (%i1) load(itensor);
      (%o1)      /share/tensor/itensor.lisp
      (%i2) icurvature([i,j,k],[l],m);
      Maxima encountered a Lisp error:

       Error in $ICURVATURE [or a callee]:
       $ICURVATURE [or a callee] requires less than three arguments.

      Automatically continuing.
      To reenable the Lisp debugger set *debugger-hook* to nil.

  However, if `icurvature` is entered in noun form, it can be evaluated using `evundiff`:

      (%i3) ishow('icurvature([i,j,k],[l],m))$
                                               l
      (%t3)                          icurvature
                                               i j k,m
      (%i4) ishow(evundiff(%))$
                   l              l         %1           l           %1
      (%t4) - ichr2        - ichr2     ichr2      - ichr2       ichr2
                   i k,j m        %1 j      i k,m        %1 j,m      i k

                      l              l         %1           l           %1
               + ichr2        + ichr2     ichr2      + ichr2       ichr2
                      i j,k m        %1 k      i j,m        %1 k,m      i j

  Note: In earlier versions of Maxima, derivative forms of the Christoffel-symbols also could not be evaluated. This has been fixed now, so `evundiff` is no longer necessary for expressions like this:

      (%i5) imetric(g);
      (%o5)                                done
      (%i6) ishow(ichr2([i,j],[k],l))$
             k %3
            g     (g         - g         + g        )
                    j %3,i l    i j,%3 l    i %3,j l
      (%t6) -----------------------------------------
                                2

                               k %3
                              g     (g       - g       + g      )
                               ,l     j %3,i    i j,%3    i %3,j
                            + -----------------------------------
                                               2

Function: flush (expr, tensor_1, tensor_2, ...)

  Set to zero, in expr, all occurrences of the tensor_i that have no derivative indices.

Function: flushd (expr, tensor_1, tensor_2, ...)

  Set to zero, in expr, all occurrences of the tensor_i that have derivative indices.

Function: flushnd (expr, tensor, n)

  Set to zero, in expr, all occurrences of the differentiated object tensor that have n or more derivative indices as the following example demonstrates.

      (%i1) load(itensor);
      (%o1)      /share/tensor/itensor.lisp
      (%i2) ishow(a([i],[J,r],k,r)+a([i],[j,r,s],k,r,s))$
                                      J r      j r s
      (%t2)                          a      + a
                                      i,k r    i,k r s
      (%i3) ishow(flushnd(%,a,3))$
                                           J r
      (%t3)                               a
                                           i,k r

Function: coord (tensor_1, tensor_2, ...)

  Gives tensor_i the coordinate differentiation property that the derivative of contravariant vector whose name is one of the tensor_i yields a Kronecker delta. For example, if `coord(x)` has been done then `idiff(x([],[i]),j)` gives `kdelta([i],[j])`. `coord` is a list of all indexed objects having this property.

Function: remcoord (tensor_1, tensor_2, ...)
Function: remcoord (all)

  Removes the coordinate differentiation property from the `tensor_i` that was established by the function `coord`. `remcoord(all)` removes this property from all indexed objects.

Function: makebox (expr)

  Display expr in the same manner as `show`; however, any tensor d'Alembertian occurring in expr will be indicated using the symbol `[]`. For example, `[]p([m],[n])` represents `g([],[i,j])*p([m],[n],i,j)`.

Function: conmetderiv (expr, tensor)

  Simplifies expressions containing ordinary derivatives of both covariant and contravariant forms of the metric tensor (the current restriction). For example, `conmetderiv` can relate the derivative of the contravariant metric tensor with the Christoffel symbols as seen from the following:

      (%i1) load(itensor);
      (%o1)      /share/tensor/itensor.lisp
      (%i2) ishow(g([],[a,b],c))$
                                            a b
      (%t2)                                g
                                            ,c
      (%i3) ishow(conmetderiv(%,g))$
                               %1 b      a       %1 a      b
      (%t3)                 - g     ichr2     - g     ichr2
                                         %1 c              %1 c

Function: simpmetderiv (expr)
Function: simpmetderiv (expr[, stop])

  Simplifies expressions containing products of the derivatives of the metric tensor. Specifically, `simpmetderiv` recognizes two identities:

         ab        ab           ab                 a
        g   g   + g   g     = (g   g  )   = (kdelta )   = 0
         ,d  bc        bc,d         bc ,d          c ,d

  hence

         ab          ab
        g   g   = - g   g
         ,d  bc          bc,d

  and

        ab          ab
       g   g     = g   g
        ,j  ab,i    ,i  ab,j

  which follows from the symmetries of the Christoffel symbols.

  The `simpmetderiv` function takes one optional parameter which, when present, causes the function to stop after the first successful substitution in a product expression. The `simpmetderiv` function also makes use of the global variable flipflag which determines how to apply a "canonical" ordering to the product indices.

  Put together, these capabilities can be used to achieve powerful simplifications that are difficult or impossible to accomplish otherwise. This is demonstrated through the following example that explicitly uses the partial simplification features of `simpmetderiv` to obtain a contractible expression:

      (%i1) load(itensor);
      (%o1)      /share/tensor/itensor.lisp
      (%i2) imetric(g);
      (%o2)                                done
      (%i3) ishow(g([],[a,b])*g([],[b,c])*g([a,b],[],d)*g([b,c],[],e))$
                                   a b  b c
      (%t3)                       g    g    g      g
                                             a b,d  b c,e
      (%i4) ishow(canform(%))$

      errexp1 has improper indices
       -- an error.  Quitting.  To debug this try debugmode(true);
      (%i5) ishow(simpmetderiv(%))$
                                   a b  b c
      (%t5)                       g    g    g      g
                                             a b,d  b c,e
      (%i6) flipflag:not flipflag;
      (%o6)                                true
      (%i7) ishow(simpmetderiv(%th(2)))$
                                     a b  b c
      (%t7)                         g    g    g    g
                                     ,d   ,e   a b  b c
      (%i8) flipflag:not flipflag;
      (%o8)                                false
      (%i9) ishow(simpmetderiv(%th(2),stop))$
                                     a b  b c
      (%t9)                       - g    g    g      g
                                          ,e   a b,d  b c
      (%i10) ishow(contract(%))$
                                          b c
      (%t10)                           - g    g
                                          ,e   c b,d

  See also `weyl.dem` for an example that uses `simpmetderiv` and `conmetderiv` together to simplify contractions of the Weyl tensor.

Function: flush1deriv (expr, tensor)

  Set to zero, in `expr`, all occurrences of `tensor` that have exactly one derivative index.

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### 28.2.4 Tensors in curved spaces

Function: imetric (g)
System variable: imetric

  Specifies the metric by assigning the variable `imetric:g` in addition, the contraction properties of the metric g are set up by executing the commands `defcon(g),defcon(g,g,kdelta)`. The variable `imetric` (unbound by default), is bound to the metric, assigned by the `imetric(g)` command.

Function: idim (n)

  Sets the dimensions of the metric. Also initializes the antisymmetry properties of the Levi-Civita symbols for the given dimension.

Function: ichr1 ([i, j, k])

  Yields the Christoffel symbol of the first kind via the definition

             (g      + g      - g     )/2 .
               ik,j     jk,i     ij,k

  To evaluate the Christoffel symbols for a particular metric, the variable `imetric` must be assigned a name as in the example under `chr2`.

Function: ichr2 ([i, j], [k])

  Yields the Christoffel symbol of the second kind defined by the relation

                             ks
         ichr2([i,j],[k]) = g    (g      + g      - g     )/2
                                   is,j     js,i     ij,s

Function: icurvature ([i, j, k], [h])

  Yields the Riemann curvature tensor in terms of the Christoffel symbols of the second kind (`ichr2`). The following notation is used:

                     h             h            h         %1         h
           icurvature     = - ichr2      - ichr2     ichr2    + ichr2
                     i j k         i k,j        %1 j      i k        i j,k
                                     h          %1
                              + ichr2      ichr2
                                     %1 k       i j

Function: covdiff (expr, v_1, v_2, ...)

  Yields the covariant derivative of expr with respect to the variables v_i in terms of the Christoffel symbols of the second kind (`ichr2`). In order to evaluate these, one should use `ev(expr,ichr2)`.

      (%i1) load(itensor);
      (%o1)      /share/tensor/itensor.lisp
      (%i2) entertensor()$
      Enter tensor name: a;
      Enter a list of the covariant indices: [i,j];
      Enter a list of the contravariant indices: [k];
      Enter a list of the derivative indices: [];
                                            k
      (%t2)                                a
                                            i j
      (%i3) ishow(covdiff(%,s))$
                   k         %1     k         %1     k            k     %1
      (%t3)     - a     ichr2    - a     ichr2    + a      + ichr2     a
                   i %1      j s    %1 j      i s    i j,s        %1 s  i j
      (%i4) imetric:g;
      (%o4)                                  g
      (%i5) ishow(ev(%th(2),ichr2))$
               %1 %4  k
              g      a     (g       - g       + g      )
                      i %1   s %4,j    j s,%4    j %4,s
      (%t5) - ------------------------------------------
                                  2
          %1 %3  k
         g      a     (g       - g       + g      )
                 %1 j   s %3,i    i s,%3    i %3,s
       - ------------------------------------------
                             2
          k %2  %1
         g     a    (g        - g        + g       )
                i j   s %2,%1    %1 s,%2    %1 %2,s     k
       + ------------------------------------------- + a
                              2                         i j,s
      (%i6)

Function: lorentz_gauge (expr)

  Imposes the Lorentz condition by substituting 0 for all indexed objects in expr that have a derivative index identical to a contravariant index.

Function: igeodesic_coords (expr, name)

  Causes undifferentiated Christoffel symbols and first derivatives of the metric tensor vanish in expr. The name in the `igeodesic_coords` function refers to the metric name (if it appears in expr) while the connection coefficients must be called with the names `ichr1` and/or `ichr2`. The following example demonstrates the verification of the cyclic identity satisfied by the Riemann curvature tensor using the `igeodesic_coords` function.

      (%i1) load(itensor);
      (%o1)      /share/tensor/itensor.lisp
      (%i2) ishow(icurvature([r,s,t],[u]))$
                   u            u         %1         u            u         %1
      (%t2) - ichr2      - ichr2     ichr2    + ichr2      + ichr2     ichr2
                   r t,s        %1 s      r t        r s,t        %1 t      r s
      (%i3) ishow(igeodesic_coords(%,ichr2))$
                                       u            u
      (%t3)                       ichr2      - ichr2
                                       r s,t        r t,s
      (%i4) ishow(igeodesic_coords(icurvature([r,s,t],[u]),ichr2)+
                  igeodesic_coords(icurvature([s,t,r],[u]),ichr2)+
                  igeodesic_coords(icurvature([t,r,s],[u]),ichr2))$
                   u            u            u            u            u
      (%t4) - ichr2      + ichr2      + ichr2      - ichr2      - ichr2
                   t s,r        t r,s        s t,r        s r,t        r t,s

                                                                        u
                                                                 + ichr2
                                                                        r s,t
      (%i5) canform(%);
      (%o5)                                  0

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### 28.2.5 Moving frames

Maxima now has the ability to perform calculations using moving frames. These can be orthonormal frames (tetrads, vielbeins) or an arbitrary frame.

To use frames, you must first set `iframe_flag` to `true`. This causes the Christoffel-symbols, `ichr1` and `ichr2`, to be replaced by the more general frame connection coefficients `icc1` and `icc2` in calculations. Speficially, the behavior of `covdiff` and `icurvature` is changed.

The frame is defined by two tensors: the inverse frame field (`ifri`, the dual basis tetrad), and the frame metric `ifg`. The frame metric is the identity matrix for orthonormal frames, or the Lorentz metric for orthonormal frames in Minkowski spacetime. The inverse frame field defines the frame base (unit vectors). Contraction properties are defined for the frame field and the frame metric.

When `iframe_flag` is true, many `itensor` expressions use the frame metric `ifg` instead of the metric defined by `imetric` for raising and lowerind indices.

IMPORTANT: Setting the variable `iframe_flag` to `true` does NOT undefine the contraction properties of a metric defined by a call to `defcon` or `imetric`. If a frame field is used, it is best to define the metric by assigning its name to the variable `imetric` and NOT invoke the `imetric` function.

Maxima uses these two tensors to define the frame coefficients (`ifc1` and `ifc2`) which form part of the connection coefficients (`icc1` and `icc2`), as the following example demonstrates:

    (%i1) load(itensor);
    (%o1)      /share/tensor/itensor.lisp
    (%i2) iframe_flag:true;
    (%o2)                                true
    (%i3) ishow(covdiff(v([],[i]),j))$
                                   i        i     %1
    (%t3)                         v   + icc2     v
                                   ,j       %1 j
    (%i4) ishow(ev(%,icc2))$
                            %1      i           i        i
    (%t4)                  v   (ifc2     + ichr2    ) + v
                                    %1 j        %1 j     ,j
    (%i5) ishow(ev(%,ifc2))$
                %1    i %2
               v   ifg     (ifb        - ifb        + ifb       )
                               j %2 %1      %2 %1 j      %1 j %2     i
    (%t5)      -------------------------------------------------- + v
                                       2                             ,j
    (%i6) ishow(ifb([a,b,c]))$
                           %5    %4
    (%t6)               ifr   ifr   (ifri        - ifri       )
                           a     b       c %4,%5       c %5,%4

An alternate method is used to compute the frame bracket (`ifb`) if the `iframe_bracket_form` flag is set to `false`:

    (%i8) block([iframe_bracket_form:false],ishow(ifb([a,b,c])))$
                           %7    %6        %6      %7
    (%t8)              (ifr   ifr     - ifr     ifr  ) ifri
                           a     b,%7      a,%7    b       c %6

Function: iframes ()

  Since in this version of Maxima, contraction identities for `ifr` and `ifri` are always defined, as is the frame bracket (`ifb`), this function does nothing.

Variable: ifb

  The frame bracket. The contribution of the frame metric to the connection coefficients is expressed using the frame bracket:

                - ifb      + ifb      + ifb
                     c a b      b c a      a b c
      ifc1    = --------------------------------
          abc                  2

  The frame bracket itself is defined in terms of the frame field and frame metric. Two alternate methods of computation are used depending on the value of `frame_bracket_form`. If true (the default) or if the `itorsion_flag` is `true`:

                d      e                                      f
      ifb =  ifr    ifr   (ifri      - ifri      - ifri    itr   )
         abc    b      c       a d,e       a e,d       a f    d e

  Otherwise:

                   e      d        d      e
      ifb    = (ifr    ifr    - ifr    ifr   ) ifri
         abc       b      c,e      b,e    c        a d

Variable: icc1

  Connection coefficients of the first kind. In `itensor`, defined as

      icc1    = ichr1    - ikt1    - inmc1
          abc        abc       abc        abc

  In this expression, if `iframe_flag` is true, the Christoffel-symbol `ichr1` is replaced with the frame connection coefficient `ifc1`. If `itorsion_flag` is `false`, `ikt1` will be omitted. It is also omitted if a frame base is used, as the torsion is already calculated as part of the frame bracket. Lastly, of `inonmet_flag` is `false`, `inmc1` will not be present.

Variable: icc2

  Connection coefficients of the second kind. In `itensor`, defined as

          c         c        c         c
      icc2   = ichr2   - ikt2   - inmc2
          ab        ab       ab        ab

  In this expression, if `iframe_flag` is true, the Christoffel-symbol `ichr2` is replaced with the frame connection coefficient `ifc2`. If `itorsion_flag` is `false`, `ikt2` will be omitted. It is also omitted if a frame base is used, as the torsion is already calculated as part of the frame bracket. Lastly, of `inonmet_flag` is `false`, `inmc2` will not be present.

Variable: ifc1

  Frame coefficient of the first kind (also known as Ricci-rotation coefficients.) This tensor represents the contribution of the frame metric to the connection coefficient of the first kind. Defined as:

                - ifb      + ifb      + ifb
                     c a b      b c a      a b c
      ifc1    = --------------------------------
          abc                   2

Variable: ifc2

  Frame coefficient of the first kind. This tensor represents the contribution of the frame metric to the connection coefficient of the first kind. Defined as a permutation of the frame bracket (`ifb`) with the appropriate indices raised and lowered as necessary:

          c       cd
      ifc2   = ifg   ifc1
          ab             abd

Variable: ifr

  The frame field. Contracts with the inverse frame field (`ifri`) to form the frame metric (`ifg`).

Variable: ifri

  The inverse frame field. Specifies the frame base (dual basis vectors). Along with the frame metric, it forms the basis of all calculations based on frames.

Variable: ifg

  The frame metric. Defaults to `kdelta`, but can be changed using `components`.

Variable: ifgi

  The inverse frame metric. Contracts with the frame metric (`ifg`) to `kdelta`.

Option variable: iframe_bracket_form

  Default value: `true`

  Specifies how the frame bracket (`ifb`) is computed.

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### 28.2.6 Torsion and nonmetricity

Maxima can now take into account torsion and nonmetricity. When the flag `itorsion_flag` is set to `true`, the contribution of torsion is added to the connection coefficients. Similarly, when the flag `inonmet_flag` is true, nonmetricity components are included.

Variable: inm

  The nonmetricity vector. Conformal nonmetricity is defined through the covariant derivative of the metric tensor. Normally zero, the metric tensor's covariant derivative will evaluate to the following when `inonmet_flag` is set to `true`:

      g     =- g  inm
       ij;k     ij   k

Variable: inmc1

  Covariant permutation of the nonmetricity vector components. Defined as

                 g   inm  - inm  g   - g   inm
                  ab    c      a  bc    ac    b
      inmc1    = ------------------------------
           abc                 2

  (Substitute `ifg` in place of `g` if a frame metric is used.)

Variable: inmc2

  Contravariant permutation of the nonmetricity vector components. Used in the connection coefficients if `inonmet_flag` is `true`. Defined as:

                            c         c         cd
                -inm  kdelta  - kdelta  inm  + g   inm  g
           c        a       b         a    b          d  ab
      inmc2   = -------------------------------------------
           ab                        2

  (Substitute `ifg` in place of `g` if a frame metric is used.)

Variable: ikt1

  Covariant permutation of the torsion tensor (also known as contorsion). Defined as:

                        d           d       d
                -g   itr  - g    itr   - itr   g
                  ad    cb    bd    ca      ab  cd
      ikt1    = ----------------------------------
          abc                   2

  (Substitute `ifg` in place of `g` if a frame metric is used.)

Variable: ikt2

  Contravariant permutation of the torsion tensor (also known as contorsion). Defined as:

          c     cd
      ikt2   = g   ikt1
          ab           abd

  (Substitute `ifg` in place of `g` if a frame metric is used.)

Variable: itr

  The torsion tensor. For a metric with torsion, repeated covariant differentiation on a scalar function will not commute, as demonstrated by the following example:

      (%i1) load(itensor);
      (%o1)      /share/tensor/itensor.lisp
      (%i2) imetric:g;
      (%o2)                                  g
      (%i3) covdiff(covdiff(f([],[]),i),j)-covdiff(covdiff(f([],[]),j),i)$
      (%i4) ishow(%)$
                                         %4              %2
      (%t4)                    f    ichr2    - f    ichr2
                                ,%4      j i    ,%2      i j
      (%i5) canform(%);
      (%o5)                                  0
      (%i6) itorsion_flag:true;
      (%o6)                                true
      (%i7) covdiff(covdiff(f([],[]),i),j)-covdiff(covdiff(f([],[]),j),i)$
      (%i8) ishow(%)$
                                 %8             %6
      (%t8)             f    icc2    - f    icc2    - f     + f
                         ,%8     j i    ,%6     i j    ,j i    ,i j
      (%i9) ishow(canform(%))$
                                         %1             %1
      (%t9)                     f    icc2    - f    icc2
                                 ,%1     j i    ,%1     i j
      (%i10) ishow(canform(ev(%,icc2)))$
                                         %1             %1
      (%t10)                    f    ikt2    - f    ikt2
                                 ,%1     i j    ,%1     j i
      (%i11) ishow(canform(ev(%,ikt2)))$
                            %2 %1                    %2 %1
      (%t11)          f    g      ikt1       - f    g      ikt1
                       ,%2            i j %1    ,%2            j i %1
      (%i12) ishow(factor(canform(rename(expand(ev(%,ikt1))))))$
                                 %3 %2            %1       %1
                           f    g      g      (itr    - itr   )
                            ,%3         %2 %1     j i      i j
      (%t12)               ------------------------------------
                                            2
      (%i13) decsym(itr,2,1,[anti(all)],[]);
      (%o13)                               done
      (%i14) defcon(g,g,kdelta);
      (%o14)                               done
      (%i15) subst(g,nounify(g),%th(3))$
      (%i16) ishow(canform(contract(%)))$
                                                 %1
      (%t16)                           - f    itr
                                          ,%1    i j

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### 28.2.7 Exterior algebra

The `itensor` package can perform operations on totally antisymmetric covariant tensor fields. A totally antisymmetric tensor field of rank (0,L) corresponds with a differential L-form. On these objects, a multiplication operation known as the exterior product, or wedge product, is defined.

Unfortunately, not all authors agree on the definition of the wedge product. Some authors prefer a definition that corresponds with the notion of antisymmetrization: in these works, the wedge product of two vector fields, for instance, would be defined as

                a a  - a a
                 i j    j i
     a  /\ a  = -----------
      i     j        2

More generally, the product of a p-form and a q-form would be defined as

                           1     k1..kp l1..lq
    A       /\ B       = ------ D              A       B
     i1..ip     j1..jq   (p+q)!  i1..ip j1..jq  k1..kp  l1..lq

where `D` stands for the Kronecker-delta.

Other authors, however, prefer a "geometric" definition that corresponds with the notion of the volume element:

    a  /\ a  = a a  - a a
     i     j    i j    j i

and, in the general case

                           1    k1..kp l1..lq
    A       /\ B       = ----- D              A       B
     i1..ip     j1..jq   p! q!  i1..ip j1..jq  k1..kp  l1..lq

Since `itensor` is a tensor algebra package, the first of these two definitions appears to be the more natural one. Many applications, however, utilize the second definition. To resolve this dilemma, a flag has been implemented that controls the behavior of the wedge product: if `igeowedge_flag` is `false` (the default), the first, "tensorial" definition is used, otherwise the second, "geometric" definition will be applied.

Operator: "~"

  The wedge product operator is denoted by the tilde `~`. This is a binary operator. Its arguments should be expressions involving scalars, covariant tensors of rank one, or covariant tensors of rank `l` that have been declared antisymmetric in all covariant indices.

  The behavior of the wedge product operator is controlled by the `igeowedge_flag` flag, as in the following example:

      (%i1) load(itensor);
      (%o1)      /share/tensor/itensor.lisp
      (%i2) ishow(a([i])~b([j]))$
                                       a  b  - b  a
                                        i  j    i  j
      (%t2)                            -------------
                                             2
      (%i3) decsym(a,2,0,[anti(all)],[]);
      (%o3)                                done
      (%i4) ishow(a([i,j])~b([k]))$
                                a    b  + b  a    - a    b
                                 i j  k    i  j k    i k  j
      (%t4)                     ---------------------------
                                             3
      (%i5) igeowedge_flag:true;
      (%o5)                                true
      (%i6) ishow(a([i])~b([j]))$
      (%t6)                            a  b  - b  a
                                        i  j    i  j
      (%i7) ishow(a([i,j])~b([k]))$
      (%t7)                     a    b  + b  a    - a    b
                                 i j  k    i  j k    i k  j

Operator: "|"

  The vertical bar `|` denotes the "contraction with a vector" binary operation. When a totally antisymmetric covariant tensor is contracted with a contravariant vector, the result is the same regardless which index was used for the contraction. Thus, it is possible to define the contraction operation in an index-free manner.

  In the `itensor` package, contraction with a vector is always carried out with respect to the first index in the literal sorting order. This ensures better simplification of expressions involving the `|` operator. For instance:

      (%i1) load(itensor);
      (%o1)      /share/tensor/itensor.lisp
      (%i2) decsym(a,2,0,[anti(all)],[]);
      (%o2)                                done
      (%i3) ishow(a([i,j],[])|v)$
                                          %1
      (%t3)                              v   a
                                              %1 j
      (%i4) ishow(a([j,i],[])|v)$
                                           %1
      (%t4)                             - v   a
                                               %1 j

  Note that it is essential that the tensors used with the `|` operator be declared totally antisymmetric in their covariant indices. Otherwise, the results will be incorrect.

Function: extdiff (expr, i)

  Computes the exterior derivative of expr with respect to the index i. The exterior derivative is formally defined as the wedge product of the partial derivative operator and a differential form. As such, this operation is also controlled by the setting of `igeowedge_flag`. For instance:

      (%i1) load(itensor);
      (%o1)      /share/tensor/itensor.lisp
      (%i2) ishow(extdiff(v([i]),j))$
                                        v    - v
                                         j,i    i,j
      (%t2)                             -----------
                                             2
      (%i3) decsym(a,2,0,[anti(all)],[]);
      (%o3)                                done
      (%i4) ishow(extdiff(a([i,j]),k))$
                                 a      - a      + a
                                  j k,i    i k,j    i j,k
      (%t4)                      ------------------------
                                            3
      (%i5) igeowedge_flag:true;
      (%o5)                                true
      (%i6) ishow(extdiff(v([i]),j))$
      (%t6)                             v    - v
                                         j,i    i,j
      (%i7) ishow(extdiff(a([i,j]),k))$
      (%t7)                      a      - a      + a
                                  j k,i    i k,j    i j,k

Function: hodge (expr)

  Compute the Hodge-dual of expr. For instance:

      (%i1) load(itensor);
      (%o1)      /share/tensor/itensor.lisp
      (%i2) imetric(g);
      (%o2)                            done
      (%i3) idim(4);
      (%o3)                            done
      (%i4) icounter:100;
      (%o4)                             100
      (%i5) decsym(A,3,0,[anti(all)],[])$

      (%i6) ishow(A([i,j,k],[]))$
      (%t6)                           A
                                       i j k
      (%i7) ishow(canform(hodge(%)))$
                                %1 %2 %3 %4
                     levi_civita            g        A
                                             %1 %102  %2 %3 %4
      (%t7)          -----------------------------------------
                                         6
      (%i8) ishow(canform(hodge(%)))$
                       %1 %2 %3 %8            %4 %5 %6 %7
      (%t8) levi_civita            levi_civita            g        g
                                                           %1 %106  %2 %107
                                                  g        g      A        /6
                                                   %3 %108  %4 %8  %5 %6 %7
      (%i9) lc2kdt(%)$

      (%i10) %,kdelta$

      (%i11) ishow(canform(contract(expand(%))))$
      (%t11)                     - A
                                    %106 %107 %108

Option variable: igeowedge_flag

  Default value: `false`

  Controls the behavior of the wedge product and exterior derivative. When set to `false` (the default), the notion of differential forms will correspond with that of a totally antisymmetric covariant tensor field. When set to `true`, differential forms will agree with the notion of the volume element.

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### 28.2.8 Exporting TeX expressions

The `itensor` package provides limited support for exporting tensor expressions to TeX. Since `itensor` expressions appear as function calls, the regular Maxima `tex` command will not produce the expected output. You can try instead the `tentex` command, which attempts to translate tensor expressions into appropriately indexed TeX objects.

Function: tentex (expr)

  To use the `tentex` function, you must first load `tentex`, as in the following example:

      (%i1) load(itensor);
      (%o1)      /share/tensor/itensor.lisp
      (%i2) load(tentex);
      (%o2)       /share/tensor/tentex.lisp
      (%i3) idummyx:m;
      (%o3)                                  m
      (%i4) ishow(icurvature([j,k,l],[i]))$
                  m1       i           m1       i           i            i
      (%t4)  ichr2    ichr2     - ichr2    ichr2     - ichr2      + ichr2
                  j k      m1 l        j l      m1 k        j l,k        j k,l
      (%i5) tentex(%)$
      $$\Gamma_{j\,k}^{m_1}\,\Gamma_{l\,m_1}^{i}-\Gamma_{j\,l}^{m_1}\,
       \Gamma_{k\,m_1}^{i}-\Gamma_{j\,l,k}^{i}+\Gamma_{j\,k,l}^{i}$$

  Note the use of the `idummyx` assignment, to avoid the appearance of the percent sign in the TeX expression, which may lead to compile errors.

  NB: This version of the `tentex` function is somewhat experimental.

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### 28.2.9 Interfacing with ctensor

The `itensor` package has the ability to generate Maxima code that can then be executed in the context of the `ctensor` package. The function that performs this task is `ic_convert`.

Function: ic_convert (eqn)

  Converts the `itensor` equation eqn to a `ctensor` assignment statement. Implied sums over dummy indices are made explicit while indexed objects are transformed into arrays (the array subscripts are in the order of covariant followed by contravariant indices of the indexed objects). The derivative of an indexed object will be replaced by the noun form of `diff` taken with respect to `ct_coords` subscripted by the derivative index. The Christoffel symbols `ichr1` and `ichr2` will be translated to `lcs` and `mcs`, respectively and if `metricconvert` is `true` then all occurrences of the metric with two covariant (contravariant) indices will be renamed to `lg` (`ug`). In addition, `do` loops will be introduced summing over all free indices so that the transformed assignment statement can be evaluated by just doing `ev`. The following examples demonstrate the features of this function.

      (%i1) load(itensor);
      (%o1)      /share/tensor/itensor.lisp
      (%i2) eqn:ishow(t([i,j],[k])=f([],[])*g([l,m],[])*a([],[m],j)*b([i],[l,k]))$
                                   k        m   l k
      (%t2)                       t    = f a   b    g
                                   i j      ,j  i    l m
      (%i3) ic_convert(eqn);
      (%o3) for i thru dim do (for j thru dim

      do (for k thru dim do t        : f sum(sum(diff(a , ct_coords ) b
                             i, j, k                   m           j   i, l, k

       g    , l, 1, dim), m, 1, dim)))
        l, m
      (%i4) imetric(g);
      (%o4)                                done
      (%i5) metricconvert:true;
      (%o5)                                true
      (%i6) ic_convert(eqn);
      (%o6) for i thru dim do (for j thru dim

      do (for k thru dim do t        : f sum(sum(diff(a , ct_coords ) b
                             i, j, k                   m           j   i, l, k

       lg    , l, 1, dim), m, 1, dim)))
         l, m

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### 28.2.10 Reserved words

The following Maxima words are used by the `itensor` package internally and should not be redefined:

      Keyword    Comments
      ------------------------------------------
      indices2() Internal version of indices()
      conti      Lists contravariant indices
      covi       Lists covariant indices of a indexed object
      deri       Lists derivative indices of an indexed object
      name       Returns the name of an indexed object
      concan
      irpmon
      lc0
      _lc2kdt0
      _lcprod
      _extlc

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