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## 45 Package cartan

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### 45.1 Introduction to cartan

`cartan` is a package which implements the calculus of differential forms. Included are facilities for computing exterior products, contraction of vectors and forms, exterior derivatives, and Lie derivatives. The differential forms are given in terms of the natural basis forms derived from the coordinates on the manifold. Type `load(cartan);` to load this package. For a demonstration, type `demo(cartan);`

`cartan` was implemented by F.B. Estabrook and H.D. Wahlquist.

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### 45.2 Functions and Variables for cartan

Function: init_cartan ([x_1, …, x_m]) ¶

  Initializes some global variables for the package. The argument, a Maxima list, is an ordered list of coordinates. The coordinate list is stored as the value of the variable `cartan_coords`. The value of the variable `cartan_basis` is a list of the basis 1-forms. The dimension is stored as the value of the variable `cartan_dim`.

    (%i1) load("cartan")$
    (%i2) init_cartan([x,y,z]);
    (%o2)                            [dx, dy, dz]
    (%i3) cartan_basis;
    (%o3)                            [dx, dy, dz]
    (%i4) cartan_dim;
    (%o4)                                  3

Infix Operator: ~ ¶

  The tilde “~” is an infix operator which denotes the exterior (wedge) product. The canonical ordering of the products is determined by the order in which the coordinates were specified. For example, if the coordinate list is specified as `[x,y,z]`, then the two-form `dy~dx` simplifies to `-dx~dy`.

    (%i1) init_cartan([x,y,z]);
    (%o1)                            [dx, dy, dz]
    (%i2) vr:[v1,v2,v3];
    (%o2)                            [v1, v2, v3]
    (%i3) f1:x~dy+z~dz;
    (%o3)                             z dz + x dy
    (%i4) f2:dz~f1;
    (%o4)                              - x dy dz

Infix Operator: | ¶

  The bar “|” is an infix operator which defines the contraction of a vector on a form. The vector should be given on the left.

Function: ext_diff (form) ¶

  The exterior derivative operator. It takes one argument, which should be a differential form.

Function: lie_diff (vector1,[vector2,form]) ¶

  The Lie derivative operator. The first argument is a vector field. The second argument may be either a vector field or a differential form.

(Note: display2d is set to false in the following example.)

    (%i1) init_cartan([x,y,z]);
    (%o1) [dx,dy,dz]
    (%i2) vr:[v1,v2,v3];
    (%o2) [v1,v2,v3]
    (%i3) f1:x ~ dy+z ~ dz;
    (%o3) z*dz+x*dy
    (%i4) vr | f1;
    (%o4) v3*z+v2*x
    (%i5) f2:dz ~ f1;
    (%o5) -(x*dy*dz)
    (%i6) ext_diff(f1);
    (%o6) dx*dy
    (%i7) ext_diff(f3(x,y,z) ~ dx);
    (%o7) -('diff(f3(x,y,z),z,1)*dx*dz)-'diff(f3(x,y,z),y,1)*dx*dy
    (%i8) depends([v1,v2,v3],[x,y,z]);
    (%o8) [v1(x,y,z),v2(x,y,z),v3(x,y,z)]
    (%i9) lie_diff(vr,f2);
    (%o9) -('diff(v3,z,1)*x*dy*dz)-'diff(v2,y,1)*x*dy*dz-v1*dy*dz
                                   -'diff(v2,x,1)*x*dx*dz+'diff(v3,x,1)*x*dx*dy
    (%i10) depends([w1,w2,w3],[x,y,z]);
    (%o10) [w1(x,y,z),w2(x,y,z),w3(x,y,z)]
    (%i11) wr:[w1,w2,w3];
    (%o11) [w1,w2,w3]
    (%i12) lie_diff(vr,wr);
    (%o12) [-('diff(v1,z,1)*w3)-'diff(v1,y,1)*w2+v3*'diff(w1,z,1)+v2*'diff(w1,y,1)
                               +v1*'diff(w1,x,1)-'diff(v1,x,1)*w1,
            -('diff(v2,z,1)*w3)+v3*'diff(w2,z,1)+v2*'diff(w2,y,1)+v1*'diff(w2,x,1)
                               -'diff(v2,y,1)*w2-'diff(v2,x,1)*w1,
            v3*'diff(w3,z,1)+v2*'diff(w3,y,1)+v1*'diff(w3,x,1)-'diff(v3,z,1)*w3
                            -'diff(v3,y,1)*w2-'diff(v3,x,1)*w1]

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