Multivector Fields - Manifolds

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# Multivector Fields

Let \(U\) and \(M\) be two differentiable manifolds. Given a positive integer \(p\) and a differentiable map \(\Phi: U \rightarrow M\), a multivector field of degree \(p\), or \(p\)-vector field, along \(U\) with values on \(M\) is a field along \(U\) of alternating contravariant tensors of rank \(p\) in the tangent spaces to \(M\). The standard case of a multivector field on a differentiable manifold corresponds to \(U = M\) and \(\Phi = \mathrm{Id}_M\). Other common cases are \(\Phi\) being an immersion and \(\Phi\) being a curve in \(M\) (\(U\) is then an open interval of \(\RR\)).

Two classes implement multivector fields, depending whether the manifold \(M\) is parallelizable:

- `MultivectorFieldParal` when \(M\) is parallelizable

- `MultivectorField` when \(M\) is not assumed parallelizable.

AUTHORS:

- Eric Gourgoulhon (2017): initial version

REFERENCES:

- R. L. Bishop and S. L. Goldberg (1980) [BG1980]

- C.-M. Marle (1997) [Mar1997]

class sage.manifolds.differentiable.multivectorfield.MultivectorField(vector_field_module, degree, name=None, latex_name=None)

  Bases: `TensorField`

  Multivector field with values on a generic (i.e. a priori not parallelizable) differentiable manifold.

  Given a differentiable manifold \(U\), a differentiable map \(\Phi: U \rightarrow M\) to a differentiable manifold \(M\) and a positive integer \(p\), a multivector field of degree \(p\) (or \(p\)-vector field) along \(U\) with values on \(M\supset\Phi(U)\) is a differentiable map

  \[a:\ U \longrightarrow T^{(p,0)}M\]

  (\(T^{(p,0)}M\) being the tensor bundle of type \((p,0)\) over \(M\)) such that

  \[\forall x \in U,\quad a(x) \in \Lambda^p(T_{\Phi(x)} M) ,\]

  where \(T_{\Phi(x)} M\) is the vector space tangent to \(M\) at \(\Phi(x)\) and \(\Lambda^p\) stands for the exterior power of degree \(p\) (cf. `ExtPowerFreeModule`). In other words, \(a(x)\) is an alternating contravariant tensor of degree \(p\) of the tangent vector space \(T_{\Phi(x)} M\).

  The standard case of a multivector field on a manifold \(M\) corresponds to \(U = M\) and \(\Phi = \mathrm{Id}_M\). Other common cases are \(\Phi\) being an immersion and \(\Phi\) being a curve in \(M\) (\(U\) is then an open interval of \(\RR\)).

  Note

  If \(M\) is parallelizable, the class `MultivectorFieldParal` must be used instead.

  INPUT:

  - `vector_field_module` – module \(\mathfrak{X}(U,\Phi)\) of vector fields along \(U\) with values on \(M\) via the map \(\Phi\)

  - `degree` – the degree of the multivector field (i.e. its tensor rank)

  - `name` – (default: `None`) name given to the multivector field

  - `latex_name` – (default: `None`) LaTeX symbol to denote the multivector field; if none is provided, the LaTeX symbol is set to `name`

  EXAMPLES:

  Multivector field of degree 2 on a non-parallelizable 2-dimensional manifold:

  Sage

      sage: M = Manifold(2, 'M')
      sage: U = M.open_subset('U') ; V = M.open_subset('V')
      sage: M.declare_union(U,V)   # M is the union of U and V
      sage: c_xy.<x,y> = U.chart() ; c_uv.<u,v> = V.chart()
      sage: xy_to_uv = c_xy.transition_map(c_uv, (x+y, x-y),
      ....:                         intersection_name='W',
      ....:                         restrictions1= x>0, restrictions2= u+v>0)
      sage: uv_to_xy = xy_to_uv.inverse()
      sage: W = U.intersection(V)
      sage: eU = c_xy.frame() ; eV = c_uv.frame()
      sage: a = M.multivector_field(2, name='a') ; a
      2-vector field a on the 2-dimensional differentiable manifold M
      sage: a.parent()
      Module A^2(M) of 2-vector fields on the 2-dimensional differentiable
       manifold M
      sage: a.degree()
      2

  Python

      >>> from sage.all import *
      >>> M = Manifold(Integer(2), 'M')
      >>> U = M.open_subset('U') ; V = M.open_subset('V')
      >>> M.declare_union(U,V)   # M is the union of U and V
      >>> c_xy = U.chart(names=('x', 'y',)); (x, y,) = c_xy._first_ngens(2); c_uv = V.chart(names=('u', 'v',)); (u, v,) = c_uv._first_ngens(2)
      >>> xy_to_uv = c_xy.transition_map(c_uv, (x+y, x-y),
      ...                         intersection_name='W',
      ...                         restrictions1= x>Integer(0), restrictions2= u+v>Integer(0))
      >>> uv_to_xy = xy_to_uv.inverse()
      >>> W = U.intersection(V)
      >>> eU = c_xy.frame() ; eV = c_uv.frame()
      >>> a = M.multivector_field(Integer(2), name='a') ; a
      2-vector field a on the 2-dimensional differentiable manifold M
      >>> a.parent()
      Module A^2(M) of 2-vector fields on the 2-dimensional differentiable
       manifold M
      >>> a.degree()
      2

  Setting the components of `a`:

  Sage

      sage: a[eU,0,1] = x*y^2 + 2*x
      sage: a.add_comp_by_continuation(eV, W, c_uv)
      sage: a.display(eU)
      a = (x*y^2 + 2*x) ∂/∂x∧∂/∂y
      sage: a.display(eV)
      a = (-1/4*u^3 + 1/4*u*v^2 - 1/4*v^3 + 1/4*(u^2 - 8)*v - 2*u) ∂/∂u∧∂/∂v

  [Python]

      >>> from sage.all import *
      >>> a[eU,Integer(0),Integer(1)] = x*y**Integer(2) + Integer(2)*x
      >>> a.add_comp_by_continuation(eV, W, c_uv)
      >>> a.display(eU)
      a = (x*y^2 + 2*x) ∂/∂x∧∂/∂y
      >>> a.display(eV)
      a = (-1/4*u^3 + 1/4*u*v^2 - 1/4*v^3 + 1/4*(u^2 - 8)*v - 2*u) ∂/∂u∧∂/∂v

  It is also possible to set the components while defining the 2-vector field definition, via a dictionary whose keys are the vector frames:

  Sage

      sage: a1 = M.multivector_field(2, {eU: [[0, x*y^2 + 2*x],
      ....:                                   [-x*y^2 - 2*x, 0]]}, name='a')
      sage: a1.add_comp_by_continuation(eV, W, c_uv)
      sage: a1 == a
      True

  Python

      >>> from sage.all import *
      >>> a1 = M.multivector_field(Integer(2), {eU: [[Integer(0), x*y**Integer(2) + Integer(2)*x],
      ...                                   [-x*y**Integer(2) - Integer(2)*x, Integer(0)]]}, name='a')
      >>> a1.add_comp_by_continuation(eV, W, c_uv)
      >>> a1 == a
      True

  The exterior product of two vector fields is a 2-vector field:

  Sage

      sage: a = M.vector_field({eU: [-y, x]}, name='a')
      sage: a.add_comp_by_continuation(eV, W, c_uv)
      sage: b = M.vector_field({eU: [1+x*y, x^2]}, name='b')
      sage: b.add_comp_by_continuation(eV, W, c_uv)
      sage: s = a.wedge(b) ; s
      2-vector field a∧b on the 2-dimensional differentiable manifold M
      sage: s.display(eU)
      a∧b = (-2*x^2*y - x) ∂/∂x∧∂/∂y
      sage: s.display(eV)
      a∧b = (1/2*u^3 - 1/2*u*v^2 - 1/2*v^3 + 1/2*(u^2 + 2)*v + u) ∂/∂u∧∂/∂v

  [Python]

      >>> from sage.all import *
      >>> a = M.vector_field({eU: [-y, x]}, name='a')
      >>> a.add_comp_by_continuation(eV, W, c_uv)
      >>> b = M.vector_field({eU: [Integer(1)+x*y, x**Integer(2)]}, name='b')
      >>> b.add_comp_by_continuation(eV, W, c_uv)
      >>> s = a.wedge(b) ; s
      2-vector field a∧b on the 2-dimensional differentiable manifold M
      >>> s.display(eU)
      a∧b = (-2*x^2*y - x) ∂/∂x∧∂/∂y
      >>> s.display(eV)
      a∧b = (1/2*u^3 - 1/2*u*v^2 - 1/2*v^3 + 1/2*(u^2 + 2)*v + u) ∂/∂u∧∂/∂v

  Multiplying a 2-vector field by a scalar field results in another 2-vector field:

  Sage

      sage: f = M.scalar_field({c_xy: (x+y)^2, c_uv: u^2}, name='f')
      sage: s = f*s ; s
      2-vector field f*(a∧b) on the 2-dimensional differentiable manifold M
      sage: s.display(eU)
      f*(a∧b) = (-2*x^2*y^3 - x^3 - (4*x^3 + x)*y^2 - 2*(x^4 + x^2)*y) ∂/∂x∧∂/∂y
      sage: s.display(eV)
      f*(a∧b) = (1/2*u^5 - 1/2*u^3*v^2 - 1/2*u^2*v^3 + u^3 + 1/2*(u^4 + 2*u^2)*v)
        ∂/∂u∧∂/∂v

  Python

      >>> from sage.all import *
      >>> f = M.scalar_field({c_xy: (x+y)**Integer(2), c_uv: u**Integer(2)}, name='f')
      >>> s = f*s ; s
      2-vector field f*(a∧b) on the 2-dimensional differentiable manifold M
      >>> s.display(eU)
      f*(a∧b) = (-2*x^2*y^3 - x^3 - (4*x^3 + x)*y^2 - 2*(x^4 + x^2)*y) ∂/∂x∧∂/∂y
      >>> s.display(eV)
      f*(a∧b) = (1/2*u^5 - 1/2*u^3*v^2 - 1/2*u^2*v^3 + u^3 + 1/2*(u^4 + 2*u^2)*v)
        ∂/∂u∧∂/∂v

  bracket(other)

    Return the Schouten-Nijenhuis bracket of `self` with another multivector field.

    The Schouten-Nijenhuis bracket extends the Lie bracket of vector fields (cf. `bracket()`) to multivector fields.

    Denoting by \(A^p(M)\) the \(C^k(M)\)-module of \(p\)-vector fields on the \(C^k\)-differentiable manifold \(M\) over the field \(K\) (cf. `MultivectorModule`), the Schouten-Nijenhuis bracket is a \(K\)-bilinear map

    \[\begin{split}\begin{array}{ccc} A^p(M)\times A^q(M) & \longrightarrow & A^{p+q-1}(M) \\ (a,b) & \longmapsto & [a,b] \end{array}\end{split}\]

    which obeys the following properties:

    - if \(p=0\) and \(q=0\), (i.e. \(a\) and \(b\) are two scalar fields), \([a,b]=0\)

    - if \(p=0\) (i.e. \(a\) is a scalar field) and \(q\geq 1\), \([a,b] = - \iota_{\mathrm{d}a} b\) (minus the interior product of the differential of \(a\) by \(b\))

    - if \(p=1\) (i.e. \(a\) is a vector field), \([a,b] = \mathcal{L}_a b\) (the Lie derivative of \(b\) along \(a\))

    - \([a,b] = -(-1)^{(p-1)(q-1)} [b,a]\)

    - for any multivector field \(c\) and \((a,b) \in A^p(M)\times A^q(M)\), \([a,.]\) obeys the graded Leibniz rule

    \[[a,b\wedge c] = [a,b]\wedge c + (-1)^{(p-1)q} b\wedge [a,c]\]

    - for \((a,b,c) \in A^p(M)\times A^q(M)\times A^r(M)\), the graded Jacobi identity holds:

    \[(-1)^{(p-1)(r-1)}[a,[b,c]] + (-1)^{(q-1)(p-1)}[b,[c,a]] + (-1)^{(r-1)(q-1)}[c,[a,b]] = 0\]

    Note

    There are two definitions of the Schouten-Nijenhuis bracket in the literature, which differ from each other when \(p\) is even by an overall sign. The definition adopted here is that of [Mar1997], [Kos1985] and Wikipedia article Schouten-Nijenhuis_bracket. The other definition, adopted e.g. by [Nij1955], [Lic1977] and [Vai1994], is \([a,b]' = (-1)^{p+1} [a,b]\).

    INPUT:

    - `other` – a multivector field

    OUTPUT:

    - instance of `MultivectorField` (or of `DiffScalarField` if \(p=1\) and \(q=0\)) representing the Schouten-Nijenhuis bracket \([a,b]\), where \(a\) is `self` and \(b\) is `other`

    EXAMPLES:

    Bracket of two vector fields on the 2-sphere:

    Sage

        sage: M = Manifold(2, 'S^2', start_index=1) # the sphere S^2
        sage: U = M.open_subset('U') ; V = M.open_subset('V')
        sage: M.declare_union(U,V)   # S^2 is the union of U and V
        sage: c_xy.<x,y> = U.chart() # stereographic coord. North
        sage: c_uv.<u,v> = V.chart() # stereographic coord. South
        sage: xy_to_uv = c_xy.transition_map(c_uv, (x/(x^2+y^2), y/(x^2+y^2)),
        ....:                intersection_name='W', restrictions1= x^2+y^2!=0,
        ....:                restrictions2= u^2+v^2!=0)
        sage: uv_to_xy = xy_to_uv.inverse()
        sage: W = U.intersection(V) # The complement of the two poles
        sage: e_xy = c_xy.frame() ; e_uv = c_uv.frame()
        sage: a = M.vector_field({e_xy: [y, x]}, name='a')
        sage: a.add_comp_by_continuation(e_uv, W, c_uv)
        sage: b = M.vector_field({e_xy: [x*y, x-y]}, name='b')
        sage: b.add_comp_by_continuation(e_uv, W, c_uv)
        sage: s = a.bracket(b); s
        Vector field [a,b] on the 2-dimensional differentiable manifold S^2
        sage: s.display(e_xy)
        [a,b] = (x^2 + y^2 - x + y) ∂/∂x + (-(x - 1)*y - x) ∂/∂y

    Python

        >>> from sage.all import *
        >>> M = Manifold(Integer(2), 'S^2', start_index=Integer(1)) # the sphere S^2
        >>> U = M.open_subset('U') ; V = M.open_subset('V')
        >>> M.declare_union(U,V)   # S^2 is the union of U and V
        >>> c_xy = U.chart(names=('x', 'y',)); (x, y,) = c_xy._first_ngens(2)# stereographic coord. North
        >>> c_uv = V.chart(names=('u', 'v',)); (u, v,) = c_uv._first_ngens(2)# stereographic coord. South
        >>> xy_to_uv = c_xy.transition_map(c_uv, (x/(x**Integer(2)+y**Integer(2)), y/(x**Integer(2)+y**Integer(2))),
        ...                intersection_name='W', restrictions1= x**Integer(2)+y**Integer(2)!=Integer(0),
        ...                restrictions2= u**Integer(2)+v**Integer(2)!=Integer(0))
        >>> uv_to_xy = xy_to_uv.inverse()
        >>> W = U.intersection(V) # The complement of the two poles
        >>> e_xy = c_xy.frame() ; e_uv = c_uv.frame()
        >>> a = M.vector_field({e_xy: [y, x]}, name='a')
        >>> a.add_comp_by_continuation(e_uv, W, c_uv)
        >>> b = M.vector_field({e_xy: [x*y, x-y]}, name='b')
        >>> b.add_comp_by_continuation(e_uv, W, c_uv)
        >>> s = a.bracket(b); s
        Vector field [a,b] on the 2-dimensional differentiable manifold S^2
        >>> s.display(e_xy)
        [a,b] = (x^2 + y^2 - x + y) ∂/∂x + (-(x - 1)*y - x) ∂/∂y

    For two vector fields, the bracket coincides with the Lie derivative:

    Sage

        sage: s == b.lie_derivative(a)
        True

    [Python]

        >>> from sage.all import *
        >>> s == b.lie_derivative(a)
        True

    Schouten-Nijenhuis bracket of a 2-vector field and a 1-vector field:

    Sage

        sage: c = a.wedge(b); c
        2-vector field a∧b on the 2-dimensional differentiable
         manifold S^2
        sage: s = c.bracket(a); s
        2-vector field [a∧b,a] on the 2-dimensional differentiable
         manifold S^2
        sage: s.display(e_xy)
        [a∧b,a] = (x^3 + (2*x - 1)*y^2 - x^2 + 2*x*y) ∂/∂x∧∂/∂y

    Python

        >>> from sage.all import *
        >>> c = a.wedge(b); c
        2-vector field a∧b on the 2-dimensional differentiable
         manifold S^2
        >>> s = c.bracket(a); s
        2-vector field [a∧b,a] on the 2-dimensional differentiable
         manifold S^2
        >>> s.display(e_xy)
        [a∧b,a] = (x^3 + (2*x - 1)*y^2 - x^2 + 2*x*y) ∂/∂x∧∂/∂y

    Since \(a\) is a vector field, we have in this case:

    Sage

        sage: s == - c.lie_derivative(a)
        True

    [Python]

        >>> from sage.all import *
        >>> s == - c.lie_derivative(a)
        True

    See also

    `MultivectorFieldParal.bracket()` for more examples and check of standards identities involving the Schouten-Nijenhuis bracket

  degree()

    Return the degree of `self`.

    OUTPUT: integer \(p\) such that `self` is a \(p\)-vector field

    EXAMPLES:

    Sage

        sage: M = Manifold(3, 'M')
        sage: a = M.multivector_field(2); a
        2-vector field on the 3-dimensional differentiable manifold M
        sage: a.degree()
        2
        sage: b = M.vector_field(); b
        Vector field on the 3-dimensional differentiable manifold M
        sage: b.degree()
        1

    Python

        >>> from sage.all import *
        >>> M = Manifold(Integer(3), 'M')
        >>> a = M.multivector_field(Integer(2)); a
        2-vector field on the 3-dimensional differentiable manifold M
        >>> a.degree()
        2
        >>> b = M.vector_field(); b
        Vector field on the 3-dimensional differentiable manifold M
        >>> b.degree()
        1

  interior_product(form)

    Interior product with a differential form.

    If `self` is a multivector field \(A\) of degree \(p\) and \(B\) is a differential form of degree \(q\geq p\) on the same manifold as \(A\), the interior product of \(A\) by \(B\) is the differential form \(\iota_A B\) of degree \(q-p\) defined by

    \[(\iota_A B)_{i_1\ldots i_{q-p}} = A^{k_1\ldots k_p} B_{k_1\ldots k_p i_1\ldots i_{q-p}}\]

    Note

    `A.interior_product(B)` yields the same result as `A.contract(0,..., p-1, B, 0,..., p-1)` (cf. `contract()`), but `interior_product` is more efficient, the alternating character of \(A\) being not used to reduce the computation in `contract()`

    INPUT:

    - `form` – differential form \(B\) (instance of `DiffForm`); the degree of \(B\) must be at least equal to the degree of `self`

    OUTPUT:

    - scalar field (case \(p=q\)) or `DiffForm` (case \(p<q\)) representing the interior product \(\iota_A B\), where \(A\) is `self`

    See also

    `interior_product()` for the interior product of a differential form with a multivector field

    EXAMPLES:

    Interior product of a vector field (\(p=1\)) with a 2-form (\(q=2\)) on the 2-sphere:

    Sage

        sage: M = Manifold(2, 'S^2', start_index=1) # the sphere S^2
        sage: U = M.open_subset('U') ; V = M.open_subset('V')
        sage: M.declare_union(U,V)   # S^2 is the union of U and V
        sage: c_xy.<x,y> = U.chart() # stereographic coord. North
        sage: c_uv.<u,v> = V.chart() # stereographic coord. South
        sage: xy_to_uv = c_xy.transition_map(c_uv, (x/(x^2+y^2), y/(x^2+y^2)),
        ....:                intersection_name='W', restrictions1= x^2+y^2!=0,
        ....:                restrictions2= u^2+v^2!=0)
        sage: uv_to_xy = xy_to_uv.inverse()
        sage: W = U.intersection(V) # The complement of the two poles
        sage: e_xy = c_xy.frame() ; e_uv = c_uv.frame()
        sage: a = M.vector_field({e_xy: [-y, x]}, name='a')
        sage: a.add_comp_by_continuation(e_uv, W, c_uv)
        sage: b = M.diff_form(2, name='b')
        sage: b[e_xy,1,2] = 4/(x^2+y^2+1)^2   # the standard area 2-form
        sage: b.add_comp_by_continuation(e_uv, W, c_uv)
        sage: b.display(e_xy)
        b = 4/(x^2 + y^2 + 1)^2 dx∧dy
        sage: b.display(e_uv)
        b = -4/(u^4 + v^4 + 2*(u^2 + 1)*v^2 + 2*u^2 + 1) du∧dv
        sage: s = a.interior_product(b); s
        1-form i_a b on the 2-dimensional differentiable manifold S^2
        sage: s.display(e_xy)
        i_a b = -4*x/(x^4 + y^4 + 2*(x^2 + 1)*y^2 + 2*x^2 + 1) dx
         - 4*y/(x^4 + y^4 + 2*(x^2 + 1)*y^2 + 2*x^2 + 1) dy
        sage: s.display(e_uv)
         i_a b = 4*u/(u^4 + v^4 + 2*(u^2 + 1)*v^2 + 2*u^2 + 1) du
          + 4*v/(u^4 + v^4 + 2*(u^2 + 1)*v^2 + 2*u^2 + 1) dv
        sage: s == a.contract(b)
        True

    Python

        >>> from sage.all import *
        >>> M = Manifold(Integer(2), 'S^2', start_index=Integer(1)) # the sphere S^2
        >>> U = M.open_subset('U') ; V = M.open_subset('V')
        >>> M.declare_union(U,V)   # S^2 is the union of U and V
        >>> c_xy = U.chart(names=('x', 'y',)); (x, y,) = c_xy._first_ngens(2)# stereographic coord. North
        >>> c_uv = V.chart(names=('u', 'v',)); (u, v,) = c_uv._first_ngens(2)# stereographic coord. South
        >>> xy_to_uv = c_xy.transition_map(c_uv, (x/(x**Integer(2)+y**Integer(2)), y/(x**Integer(2)+y**Integer(2))),
        ...                intersection_name='W', restrictions1= x**Integer(2)+y**Integer(2)!=Integer(0),
        ...                restrictions2= u**Integer(2)+v**Integer(2)!=Integer(0))
        >>> uv_to_xy = xy_to_uv.inverse()
        >>> W = U.intersection(V) # The complement of the two poles
        >>> e_xy = c_xy.frame() ; e_uv = c_uv.frame()
        >>> a = M.vector_field({e_xy: [-y, x]}, name='a')
        >>> a.add_comp_by_continuation(e_uv, W, c_uv)
        >>> b = M.diff_form(Integer(2), name='b')
        >>> b[e_xy,Integer(1),Integer(2)] = Integer(4)/(x**Integer(2)+y**Integer(2)+Integer(1))**Integer(2)   # the standard area 2-form
        >>> b.add_comp_by_continuation(e_uv, W, c_uv)
        >>> b.display(e_xy)
        b = 4/(x^2 + y^2 + 1)^2 dx∧dy
        >>> b.display(e_uv)
        b = -4/(u^4 + v^4 + 2*(u^2 + 1)*v^2 + 2*u^2 + 1) du∧dv
        >>> s = a.interior_product(b); s
        1-form i_a b on the 2-dimensional differentiable manifold S^2
        >>> s.display(e_xy)
        i_a b = -4*x/(x^4 + y^4 + 2*(x^2 + 1)*y^2 + 2*x^2 + 1) dx
         - 4*y/(x^4 + y^4 + 2*(x^2 + 1)*y^2 + 2*x^2 + 1) dy
        >>> s.display(e_uv)
         i_a b = 4*u/(u^4 + v^4 + 2*(u^2 + 1)*v^2 + 2*u^2 + 1) du
          + 4*v/(u^4 + v^4 + 2*(u^2 + 1)*v^2 + 2*u^2 + 1) dv
        >>> s == a.contract(b)
        True

    Example with \(p=2\) and \(q=2\):

    Sage

        sage: a = M.multivector_field(2, name='a')
        sage: a[e_xy,1,2] = x*y
        sage: a.add_comp_by_continuation(e_uv, W, c_uv)
        sage: a.display(e_xy)
        a = x*y ∂/∂x∧∂/∂y
        sage: a.display(e_uv)
        a = -u*v ∂/∂u∧∂/∂v
        sage: s = a.interior_product(b); s
        Scalar field i_a b on the 2-dimensional differentiable manifold S^2
        sage: s.display()
        i_a b: S^2 → ℝ
        on U: (x, y) ↦ 8*x*y/(x^4 + y^4 + 2*(x^2 + 1)*y^2 + 2*x^2 + 1)
        on V: (u, v) ↦ 8*u*v/(u^4 + v^4 + 2*(u^2 + 1)*v^2 + 2*u^2 + 1)

    [Python]

        >>> from sage.all import *
        >>> a = M.multivector_field(Integer(2), name='a')
        >>> a[e_xy,Integer(1),Integer(2)] = x*y
        >>> a.add_comp_by_continuation(e_uv, W, c_uv)
        >>> a.display(e_xy)
        a = x*y ∂/∂x∧∂/∂y
        >>> a.display(e_uv)
        a = -u*v ∂/∂u∧∂/∂v
        >>> s = a.interior_product(b); s
        Scalar field i_a b on the 2-dimensional differentiable manifold S^2
        >>> s.display()
        i_a b: S^2 → ℝ
        on U: (x, y) ↦ 8*x*y/(x^4 + y^4 + 2*(x^2 + 1)*y^2 + 2*x^2 + 1)
        on V: (u, v) ↦ 8*u*v/(u^4 + v^4 + 2*(u^2 + 1)*v^2 + 2*u^2 + 1)

    Some checks:

    Sage

        sage: s == a.contract(0, 1, b, 0, 1)
        True
        sage: s.restrict(U) == 2 * a[[e_xy,1,2]] * b[[e_xy,1,2]]
        True
        sage: s.restrict(V) == 2 * a[[e_uv,1,2]] * b[[e_uv,1,2]]
        True

    Python

        >>> from sage.all import *
        >>> s == a.contract(Integer(0), Integer(1), b, Integer(0), Integer(1))
        True
        >>> s.restrict(U) == Integer(2) * a[[e_xy,Integer(1),Integer(2)]] * b[[e_xy,Integer(1),Integer(2)]]
        True
        >>> s.restrict(V) == Integer(2) * a[[e_uv,Integer(1),Integer(2)]] * b[[e_uv,Integer(1),Integer(2)]]
        True

  wedge(other)

    Exterior product with another multivector field.

    INPUT:

    - `other` – another multivector field (on the same manifold)

    OUTPUT:

    - instance of `MultivectorField` representing the exterior product `self ∧ other`

    EXAMPLES:

    Exterior product of two vector fields on the 2-sphere:

    Sage

        sage: M = Manifold(2, 'S^2', start_index=1) # the sphere S^2
        sage: U = M.open_subset('U') ; V = M.open_subset('V')
        sage: M.declare_union(U,V)   # S^2 is the union of U and V
        sage: c_xy.<x,y> = U.chart() # stereographic coord. North
        sage: c_uv.<u,v> = V.chart() # stereographic coord. South
        sage: xy_to_uv = c_xy.transition_map(c_uv, (x/(x^2+y^2), y/(x^2+y^2)),
        ....:                intersection_name='W', restrictions1= x^2+y^2!=0,
        ....:                restrictions2= u^2+v^2!=0)
        sage: uv_to_xy = xy_to_uv.inverse()
        sage: W = U.intersection(V) # The complement of the two poles
        sage: e_xy = c_xy.frame() ; e_uv = c_uv.frame()
        sage: a = M.vector_field({e_xy: [y, x]}, name='a')
        sage: a.add_comp_by_continuation(e_uv, W, c_uv)
        sage: b = M.vector_field({e_xy: [x^2 + y^2, y]}, name='b')
        sage: b.add_comp_by_continuation(e_uv, W, c_uv)
        sage: c = a.wedge(b); c
        2-vector field a∧b on the 2-dimensional differentiable
         manifold S^2
        sage: c.display(e_xy)
        a∧b = (-x^3 - (x - 1)*y^2) ∂/∂x∧∂/∂y
        sage: c.display(e_uv)
        a∧b = (-v^2 + u) ∂/∂u∧∂/∂v

    Python

        >>> from sage.all import *
        >>> M = Manifold(Integer(2), 'S^2', start_index=Integer(1)) # the sphere S^2
        >>> U = M.open_subset('U') ; V = M.open_subset('V')
        >>> M.declare_union(U,V)   # S^2 is the union of U and V
        >>> c_xy = U.chart(names=('x', 'y',)); (x, y,) = c_xy._first_ngens(2)# stereographic coord. North
        >>> c_uv = V.chart(names=('u', 'v',)); (u, v,) = c_uv._first_ngens(2)# stereographic coord. South
        >>> xy_to_uv = c_xy.transition_map(c_uv, (x/(x**Integer(2)+y**Integer(2)), y/(x**Integer(2)+y**Integer(2))),
        ...                intersection_name='W', restrictions1= x**Integer(2)+y**Integer(2)!=Integer(0),
        ...                restrictions2= u**Integer(2)+v**Integer(2)!=Integer(0))
        >>> uv_to_xy = xy_to_uv.inverse()
        >>> W = U.intersection(V) # The complement of the two poles
        >>> e_xy = c_xy.frame() ; e_uv = c_uv.frame()
        >>> a = M.vector_field({e_xy: [y, x]}, name='a')
        >>> a.add_comp_by_continuation(e_uv, W, c_uv)
        >>> b = M.vector_field({e_xy: [x**Integer(2) + y**Integer(2), y]}, name='b')
        >>> b.add_comp_by_continuation(e_uv, W, c_uv)
        >>> c = a.wedge(b); c
        2-vector field a∧b on the 2-dimensional differentiable
         manifold S^2
        >>> c.display(e_xy)
        a∧b = (-x^3 - (x - 1)*y^2) ∂/∂x∧∂/∂y
        >>> c.display(e_uv)
        a∧b = (-v^2 + u) ∂/∂u∧∂/∂v

class sage.manifolds.differentiable.multivectorfield.MultivectorFieldParal(vector_field_module, degree, name=None, latex_name=None)

  Bases: `AlternatingContrTensor`, `TensorFieldParal`

  Multivector field with values on a parallelizable manifold.

  Given a differentiable manifold \(U\), a differentiable map \(\Phi: U \rightarrow M\) to a parallelizable manifold \(M\) and a positive integer \(p\), a multivector field of degree \(p\) (or \(p\)-vector field) along \(U\) with values on \(M\supset\Phi(U)\) is a differentiable map

  \[a:\ U \longrightarrow T^{(p,0)}M\]

  (\(T^{(p,0)}M\) being the tensor bundle of type \((p,0)\) over \(M\)) such that

  \[\forall x \in U,\quad a(x) \in \Lambda^p(T_{\Phi(x)} M) ,\]

  where \(T_{\Phi(x)} M\) is the vector space tangent to \(M\) at \(\Phi(x)\) and \(\Lambda^p\) stands for the exterior power of degree \(p\) (cf. `ExtPowerFreeModule`). In other words, \(a(x)\) is an alternating contravariant tensor of degree \(p\) of the tangent vector space \(T_{\Phi(x)} M\).

  The standard case of a multivector field on a manifold \(M\) corresponds to \(U = M\) and \(\Phi = \mathrm{Id}_M\). Other common cases are \(\Phi\) being an immersion and \(\Phi\) being a curve in \(M\) (\(U\) is then an open interval of \(\RR\)).

  Note

  If \(M\) is not parallelizable, the class `MultivectorField` must be used instead.

  INPUT:

  - `vector_field_module` – free module \(\mathfrak{X}(U,\Phi)\) of vector fields along \(U\) with values on \(M\) via the map \(\Phi\)

  - `degree` – the degree of the multivector field (i.e. its tensor rank)

  - `name` – (default: `None`) name given to the multivector field

  - `latex_name` – (default: `None`) LaTeX symbol to denote the multivector field; if none is provided, the LaTeX symbol is set to `name`

  EXAMPLES:

  A 2-vector field on a 4-dimensional manifold:

  Sage

      sage: M = Manifold(4, 'M')
      sage: c_txyz.<t,x,y,z> = M.chart()
      sage: a = M.multivector_field(2, name='a') ; a
      2-vector field a on the 4-dimensional differentiable manifold M
      sage: a.parent()
      Free module A^2(M) of 2-vector fields on the 4-dimensional
       differentiable manifold M

  Python

      >>> from sage.all import *
      >>> M = Manifold(Integer(4), 'M')
      >>> c_txyz = M.chart(names=('t', 'x', 'y', 'z',)); (t, x, y, z,) = c_txyz._first_ngens(4)
      >>> a = M.multivector_field(Integer(2), name='a') ; a
      2-vector field a on the 4-dimensional differentiable manifold M
      >>> a.parent()
      Free module A^2(M) of 2-vector fields on the 4-dimensional
       differentiable manifold M

  A multivector field is a tensor field of purely contravariant type:

  Sage

      sage: a.tensor_type()
      (2, 0)

  [Python]

      >>> from sage.all import *
      >>> a.tensor_type()
      (2, 0)

  It is antisymmetric, its components being `CompFullyAntiSym`:

  Sage

      sage: a.symmetries()
      no symmetry;  antisymmetry: (0, 1)
      sage: a[0,1] = 2*x
      sage: a[1,0]
      -2*x
      sage: a.comp()
      Fully antisymmetric 2-indices components w.r.t. Coordinate frame
       (M, (∂/∂t,∂/∂x,∂/∂y,∂/∂z))
      sage: type(a.comp())
      <class 'sage.tensor.modules.comp.CompFullyAntiSym'>

  Python

      >>> from sage.all import *
      >>> a.symmetries()
      no symmetry;  antisymmetry: (0, 1)
      >>> a[Integer(0),Integer(1)] = Integer(2)*x
      >>> a[Integer(1),Integer(0)]
      -2*x
      >>> a.comp()
      Fully antisymmetric 2-indices components w.r.t. Coordinate frame
       (M, (∂/∂t,∂/∂x,∂/∂y,∂/∂z))
      >>> type(a.comp())
      <class 'sage.tensor.modules.comp.CompFullyAntiSym'>

  Setting a component with repeated indices to a nonzero value results in an error:

  Sage

      sage: a[1,1] = 3
      Traceback (most recent call last):
      ...
      ValueError: by antisymmetry, the component cannot have a nonzero value
       for the indices (1, 1)
      sage: a[1,1] = 0  # OK, albeit useless
      sage: a[1,2] = 3  # OK

  [Python]

      >>> from sage.all import *
      >>> a[Integer(1),Integer(1)] = Integer(3)
      Traceback (most recent call last):
      ...
      ValueError: by antisymmetry, the component cannot have a nonzero value
       for the indices (1, 1)
      >>> a[Integer(1),Integer(1)] = Integer(0)  # OK, albeit useless
      >>> a[Integer(1),Integer(2)] = Integer(3)  # OK

  The expansion of a multivector field with respect to a given frame is displayed via the method `display()`:

  Sage

      sage: a.display() # expansion w.r.t. the default frame
      a = 2*x ∂/∂t∧∂/∂x + 3 ∂/∂x∧∂/∂y
      sage: latex(a.display()) # output for the notebook
      a = 2 \, x \frac{\partial}{\partial t }\wedge \frac{\partial}{\partial x }
       + 3 \frac{\partial}{\partial x }\wedge \frac{\partial}{\partial y }

  Python

      >>> from sage.all import *
      >>> a.display() # expansion w.r.t. the default frame
      a = 2*x ∂/∂t∧∂/∂x + 3 ∂/∂x∧∂/∂y
      >>> latex(a.display()) # output for the notebook
      a = 2 \, x \frac{\partial}{\partial t }\wedge \frac{\partial}{\partial x }
       + 3 \frac{\partial}{\partial x }\wedge \frac{\partial}{\partial y }

  Multivector fields can be added or subtracted:

  Sage

      sage: b = M.multivector_field(2)
      sage: b[0,1], b[0,2], b[0,3] = y, 2, x+z
      sage: s = a + b ; s
      2-vector field on the 4-dimensional differentiable manifold M
      sage: s.display()
      (2*x + y) ∂/∂t∧∂/∂x + 2 ∂/∂t∧∂/∂y + (x + z) ∂/∂t∧∂/∂z + 3 ∂/∂x∧∂/∂y
      sage: s = a - b ; s
      2-vector field on the 4-dimensional differentiable manifold M
      sage: s.display()
      (2*x - y) ∂/∂t∧∂/∂x - 2 ∂/∂t∧∂/∂y + (-x - z) ∂/∂t∧∂/∂z + 3 ∂/∂x∧∂/∂y

  [Python]

      >>> from sage.all import *
      >>> b = M.multivector_field(Integer(2))
      >>> b[Integer(0),Integer(1)], b[Integer(0),Integer(2)], b[Integer(0),Integer(3)] = y, Integer(2), x+z
      >>> s = a + b ; s
      2-vector field on the 4-dimensional differentiable manifold M
      >>> s.display()
      (2*x + y) ∂/∂t∧∂/∂x + 2 ∂/∂t∧∂/∂y + (x + z) ∂/∂t∧∂/∂z + 3 ∂/∂x∧∂/∂y
      >>> s = a - b ; s
      2-vector field on the 4-dimensional differentiable manifold M
      >>> s.display()
      (2*x - y) ∂/∂t∧∂/∂x - 2 ∂/∂t∧∂/∂y + (-x - z) ∂/∂t∧∂/∂z + 3 ∂/∂x∧∂/∂y

  An example of 3-vector field in \(\RR^3\) with Cartesian coordinates:

  Sage

      sage: M = Manifold(3, 'R3', latex_name=r'\RR^3', start_index=1)
      sage: c_cart.<x,y,z> = M.chart()
      sage: a = M.multivector_field(3, name='a')
      sage: a[1,2,3] = x^2+y^2+z^2  # the only independent component
      sage: a[:] # all the components are set from the previous line:
      [[[0, 0, 0], [0, 0, x^2 + y^2 + z^2], [0, -x^2 - y^2 - z^2, 0]],
       [[0, 0, -x^2 - y^2 - z^2], [0, 0, 0], [x^2 + y^2 + z^2, 0, 0]],
       [[0, x^2 + y^2 + z^2, 0], [-x^2 - y^2 - z^2, 0, 0], [0, 0, 0]]]
      sage: a.display()
      a = (x^2 + y^2 + z^2) ∂/∂x∧∂/∂y∧∂/∂z

  Python

      >>> from sage.all import *
      >>> M = Manifold(Integer(3), 'R3', latex_name=r'\RR^3', start_index=Integer(1))
      >>> c_cart = M.chart(names=('x', 'y', 'z',)); (x, y, z,) = c_cart._first_ngens(3)
      >>> a = M.multivector_field(Integer(3), name='a')
      >>> a[Integer(1),Integer(2),Integer(3)] = x**Integer(2)+y**Integer(2)+z**Integer(2)  # the only independent component
      >>> a[:] # all the components are set from the previous line:
      [[[0, 0, 0], [0, 0, x^2 + y^2 + z^2], [0, -x^2 - y^2 - z^2, 0]],
       [[0, 0, -x^2 - y^2 - z^2], [0, 0, 0], [x^2 + y^2 + z^2, 0, 0]],
       [[0, x^2 + y^2 + z^2, 0], [-x^2 - y^2 - z^2, 0, 0], [0, 0, 0]]]
      >>> a.display()
      a = (x^2 + y^2 + z^2) ∂/∂x∧∂/∂y∧∂/∂z

  Spherical components from the tensorial change-of-frame formula:

  Sage

      sage: c_spher.<r,th,ph> = M.chart(r'r:[0,+oo) th:[0,pi]:\theta ph:[0,2*pi):\phi')
      sage: spher_to_cart = c_spher.transition_map(c_cart,
      ....:                [r*sin(th)*cos(ph), r*sin(th)*sin(ph), r*cos(th)])
      sage: cart_to_spher = spher_to_cart.set_inverse(sqrt(x^2+y^2+z^2),
      ....:                              atan2(sqrt(x^2+y^2),z), atan2(y, x))
      Check of the inverse coordinate transformation:
        r == r  *passed*
        th == arctan2(r*sin(th), r*cos(th))  **failed**
        ph == arctan2(r*sin(ph)*sin(th), r*cos(ph)*sin(th))  **failed**
        x == x  *passed*
        y == y  *passed*
        z == z  *passed*
      NB: a failed report can reflect a mere lack of simplification.
      sage: a.comp(c_spher.frame()) # computation of components w.r.t. spherical frame
      Fully antisymmetric 3-indices components w.r.t. Coordinate frame
       (R3, (∂/∂r,∂/∂th,∂/∂ph))
      sage: a.comp(c_spher.frame())[1,2,3, c_spher]
      1/sin(th)
      sage: a.display(c_spher.frame())
      a = sqrt(x^2 + y^2 + z^2)/sqrt(x^2 + y^2) ∂/∂r∧∂/∂th∧∂/∂ph
      sage: a.display(c_spher.frame(), c_spher)
      a = 1/sin(th) ∂/∂r∧∂/∂th∧∂/∂ph

  [Python]

      >>> from sage.all import *
      >>> c_spher = M.chart(r'r:[0,+oo) th:[0,pi]:\theta ph:[0,2*pi):\phi', names=('r', 'th', 'ph',)); (r, th, ph,) = c_spher._first_ngens(3)
      >>> spher_to_cart = c_spher.transition_map(c_cart,
      ...                [r*sin(th)*cos(ph), r*sin(th)*sin(ph), r*cos(th)])
      >>> cart_to_spher = spher_to_cart.set_inverse(sqrt(x**Integer(2)+y**Integer(2)+z**Integer(2)),
      ...                              atan2(sqrt(x**Integer(2)+y**Integer(2)),z), atan2(y, x))
      Check of the inverse coordinate transformation:
        r == r  *passed*
        th == arctan2(r*sin(th), r*cos(th))  **failed**
        ph == arctan2(r*sin(ph)*sin(th), r*cos(ph)*sin(th))  **failed**
        x == x  *passed*
        y == y  *passed*
        z == z  *passed*
      NB: a failed report can reflect a mere lack of simplification.
      >>> a.comp(c_spher.frame()) # computation of components w.r.t. spherical frame
      Fully antisymmetric 3-indices components w.r.t. Coordinate frame
       (R3, (∂/∂r,∂/∂th,∂/∂ph))
      >>> a.comp(c_spher.frame())[Integer(1),Integer(2),Integer(3), c_spher]
      1/sin(th)
      >>> a.display(c_spher.frame())
      a = sqrt(x^2 + y^2 + z^2)/sqrt(x^2 + y^2) ∂/∂r∧∂/∂th∧∂/∂ph
      >>> a.display(c_spher.frame(), c_spher)
      a = 1/sin(th) ∂/∂r∧∂/∂th∧∂/∂ph

  As a shortcut of the above command, on can pass just the chart `c_spher` to `display`, the vector frame being then assumed to be the coordinate frame associated with the chart:

  Sage

      sage: a.display(c_spher)
      a = 1/sin(th) ∂/∂r∧∂/∂th∧∂/∂ph

  Python

      >>> from sage.all import *
      >>> a.display(c_spher)
      a = 1/sin(th) ∂/∂r∧∂/∂th∧∂/∂ph

  The exterior product of two multivector fields is performed via the method `wedge()`:

  Sage

      sage: a = M.vector_field([x*y, -z*x, y], name='A')
      sage: b = M.vector_field([y, z+y, x^2-z^2], name='B')
      sage: ab = a.wedge(b) ; ab
      2-vector field A∧B on the 3-dimensional differentiable manifold R3
      sage: ab.display()
      A∧B = (x*y^2 + 2*x*y*z) ∂/∂x∧∂/∂y + (x^3*y - x*y*z^2 - y^2) ∂/∂x∧∂/∂z
       + (x*z^3 - y^2 - (x^3 + y)*z) ∂/∂y∧∂/∂z

  [Python]

      >>> from sage.all import *
      >>> a = M.vector_field([x*y, -z*x, y], name='A')
      >>> b = M.vector_field([y, z+y, x**Integer(2)-z**Integer(2)], name='B')
      >>> ab = a.wedge(b) ; ab
      2-vector field A∧B on the 3-dimensional differentiable manifold R3
      >>> ab.display()
      A∧B = (x*y^2 + 2*x*y*z) ∂/∂x∧∂/∂y + (x^3*y - x*y*z^2 - y^2) ∂/∂x∧∂/∂z
       + (x*z^3 - y^2 - (x^3 + y)*z) ∂/∂y∧∂/∂z

  Let us check the formula relating the exterior product to the tensor product for vector fields:

  Sage

      sage: a.wedge(b) == a*b - b*a
      True

  Python

      >>> from sage.all import *
      >>> a.wedge(b) == a*b - b*a
      True

  The tensor product of a vector field and a 2-vector field is not a 3-vector field but a tensor field of type \((3,0)\) with less symmetries:

  Sage

      sage: c = a*ab ; c
      Tensor field A⊗(A∧B) of type (3,0) on the 3-dimensional differentiable
       manifold R3
      sage: c.symmetries()  # the antisymmetry is only w.r.t. the last 2 arguments:
      no symmetry;  antisymmetry: (1, 2)

  [Python]

      >>> from sage.all import *
      >>> c = a*ab ; c
      Tensor field A⊗(A∧B) of type (3,0) on the 3-dimensional differentiable
       manifold R3
      >>> c.symmetries()  # the antisymmetry is only w.r.t. the last 2 arguments:
      no symmetry;  antisymmetry: (1, 2)

  The Lie derivative of a 2-vector field is a 2-vector field:

  Sage

      sage: ab.lie_der(a)
      2-vector field on the 3-dimensional differentiable manifold R3

  Python

      >>> from sage.all import *
      >>> ab.lie_der(a)
      2-vector field on the 3-dimensional differentiable manifold R3

  bracket(other)

    Return the Schouten-Nijenhuis bracket of `self` with another multivector field.

    The Schouten-Nijenhuis bracket extends the Lie bracket of vector fields (cf. `bracket()`) to multivector fields.

    Denoting by \(A^p(M)\) the \(C^k(M)\)-module of \(p\)-vector fields on the \(C^k\)-differentiable manifold \(M\) over the field \(K\) (cf. `MultivectorModule`), the Schouten-Nijenhuis bracket is a \(K\)-bilinear map

    \[\begin{split}\begin{array}{ccc} A^p(M)\times A^q(M) & \longrightarrow & A^{p+q-1}(M) \\ (a,b) & \longmapsto & [a,b] \end{array}\end{split}\]

    which obeys the following properties:

    - if \(p=0\) and \(q=0\), (i.e. \(a\) and \(b\) are two scalar fields), \([a,b]=0\)

    - if \(p=0\) (i.e. \(a\) is a scalar field) and \(q\geq 1\), \([a,b] = - \iota_{\mathrm{d}a} b\) (minus the interior product of the differential of \(a\) by \(b\))

    - if \(p=1\) (i.e. \(a\) is a vector field), \([a,b] = \mathcal{L}_a b\) (the Lie derivative of \(b\) along \(a\))

    - \([a,b] = -(-1)^{(p-1)(q-1)} [b,a]\)

    - for any multivector field \(c\) and \((a,b) \in A^p(M)\times A^q(M)\), \([a,.]\) obeys the graded Leibniz rule

    \[[a,b\wedge c] = [a,b]\wedge c + (-1)^{(p-1)q} b\wedge [a,c]\]

    - for \((a,b,c) \in A^p(M)\times A^q(M)\times A^r(M)\), the graded Jacobi identity holds:

    \[(-1)^{(p-1)(r-1)}[a,[b,c]] + (-1)^{(q-1)(p-1)}[b,[c,a]] + (-1)^{(r-1)(q-1)}[c,[a,b]] = 0\]

    Note

    There are two definitions of the Schouten-Nijenhuis bracket in the literature, which differ from each other when \(p\) is even by an overall sign. The definition adopted here is that of [Mar1997], [Kos1985] and Wikipedia article Schouten-Nijenhuis_bracket. The other definition, adopted e.g. by [Nij1955], [Lic1977] and [Vai1994], is \([a,b]' = (-1)^{p+1} [a,b]\).

    INPUT:

    - `other` – a multivector field

    OUTPUT:

    - instance of `MultivectorFieldParal` (or of `DiffScalarField` if \(p=1\) and \(q=0\)) representing the Schouten-Nijenhuis bracket \([a,b]\), where \(a\) is `self` and \(b\) is `other`

    EXAMPLES:

    Let us consider two vector fields on a 3-dimensional manifold:

    Sage

        sage: M = Manifold(3, 'M')
        sage: X.<x,y,z> = M.chart()
        sage: a = M.vector_field([x*y+z, x+y-z, z-2*x+y], name='a')
        sage: b = M.vector_field([y+2*z-x, x^2-y+z, z-x], name='b')

    Python

        >>> from sage.all import *
        >>> M = Manifold(Integer(3), 'M')
        >>> X = M.chart(names=('x', 'y', 'z',)); (x, y, z,) = X._first_ngens(3)
        >>> a = M.vector_field([x*y+z, x+y-z, z-Integer(2)*x+y], name='a')
        >>> b = M.vector_field([y+Integer(2)*z-x, x**Integer(2)-y+z, z-x], name='b')

    and form their Schouten-Nijenhuis bracket:

    Sage

        sage: s = a.bracket(b); s
        Vector field [a,b] on the 3-dimensional differentiable manifold M
        sage: s.display()
        [a,b] = (-x^3 + (x + 3)*y - y^2 - (x + 2*y + 1)*z - 2*x) ∂/∂x
         + (2*x^2*y - x^2 + 2*x*z - 3*x) ∂/∂y
         + (-x^2 - (x - 4)*y - 3*x + 2*z) ∂/∂z

    [Python]

        >>> from sage.all import *
        >>> s = a.bracket(b); s
        Vector field [a,b] on the 3-dimensional differentiable manifold M
        >>> s.display()
        [a,b] = (-x^3 + (x + 3)*y - y^2 - (x + 2*y + 1)*z - 2*x) ∂/∂x
         + (2*x^2*y - x^2 + 2*x*z - 3*x) ∂/∂y
         + (-x^2 - (x - 4)*y - 3*x + 2*z) ∂/∂z

    Check that \([a,b]\) is actually the Lie bracket:

    Sage

        sage: f = M.scalar_field({X: x+y*z}, name='f')
        sage: s(f) == a(b(f)) - b(a(f))
        True

    Python

        >>> from sage.all import *
        >>> f = M.scalar_field({X: x+y*z}, name='f')
        >>> s(f) == a(b(f)) - b(a(f))
        True

    Check that \([a,b]\) coincides with the Lie derivative of \(b\) along \(a\):

    Sage

        sage: s == b.lie_derivative(a)
        True

    [Python]

        >>> from sage.all import *
        >>> s == b.lie_derivative(a)
        True

    Schouten-Nijenhuis bracket for \(p=0\) and \(q=1\):

    Sage

        sage: s = f.bracket(a); s
        Scalar field -i_df a on the 3-dimensional differentiable manifold M
        sage: s.display()
        -i_df a: M → ℝ
           (x, y, z) ↦ x*y - y^2 - (x + 2*y + 1)*z + z^2

    Python

        >>> from sage.all import *
        >>> s = f.bracket(a); s
        Scalar field -i_df a on the 3-dimensional differentiable manifold M
        >>> s.display()
        -i_df a: M → ℝ
           (x, y, z) ↦ x*y - y^2 - (x + 2*y + 1)*z + z^2

    Check that \([f,a] = - \iota_{\mathrm{d}f} a = - \mathrm{d}f(a)\):

    Sage

        sage: s == - f.differential()(a)
        True

    [Python]

        >>> from sage.all import *
        >>> s == - f.differential()(a)
        True

    Schouten-Nijenhuis bracket for \(p=0\) and \(q=2\):

    Sage

        sage: c = M.multivector_field(2, name='c')
        sage: c[0,1], c[0,2], c[1,2] = x+z+1, x*y+z, x-y
        sage: s = f.bracket(c); s
        Vector field -i_df c on the 3-dimensional differentiable manifold M
        sage: s.display()
        -i_df c = (x*y^2 + (x + y + 1)*z + z^2) ∂/∂x
         + (x*y - y^2 - x - z - 1) ∂/∂y + (-x*y - (x - y + 1)*z) ∂/∂z

    Python

        >>> from sage.all import *
        >>> c = M.multivector_field(Integer(2), name='c')
        >>> c[Integer(0),Integer(1)], c[Integer(0),Integer(2)], c[Integer(1),Integer(2)] = x+z+Integer(1), x*y+z, x-y
        >>> s = f.bracket(c); s
        Vector field -i_df c on the 3-dimensional differentiable manifold M
        >>> s.display()
        -i_df c = (x*y^2 + (x + y + 1)*z + z^2) ∂/∂x
         + (x*y - y^2 - x - z - 1) ∂/∂y + (-x*y - (x - y + 1)*z) ∂/∂z

    Check that \([f,c] = - \iota_{\mathrm{d}f} c\):

    Sage

        sage: s == - f.differential().interior_product(c)
        True

    [Python]

        >>> from sage.all import *
        >>> s == - f.differential().interior_product(c)
        True

    Schouten-Nijenhuis bracket for \(p=1\) and \(q=2\):

    Sage

        sage: s = a.bracket(c); s
        2-vector field [a,c] on the 3-dimensional differentiable manifold M
        sage: s.display()
        [a,c] = ((x - 1)*y - (y - 2)*z - 2*x - 1) ∂/∂x∧∂/∂y
         + ((x + 1)*y - (x + 1)*z - 3*x - 1) ∂/∂x∧∂/∂z
         + (-5*x + y - z - 2) ∂/∂y∧∂/∂z

    Python

        >>> from sage.all import *
        >>> s = a.bracket(c); s
        2-vector field [a,c] on the 3-dimensional differentiable manifold M
        >>> s.display()
        [a,c] = ((x - 1)*y - (y - 2)*z - 2*x - 1) ∂/∂x∧∂/∂y
         + ((x + 1)*y - (x + 1)*z - 3*x - 1) ∂/∂x∧∂/∂z
         + (-5*x + y - z - 2) ∂/∂y∧∂/∂z

    Again, since \(a\) is a vector field, the Schouten-Nijenhuis bracket coincides with the Lie derivative:

    Sage

        sage: s == c.lie_derivative(a)
        True

    [Python]

        >>> from sage.all import *
        >>> s == c.lie_derivative(a)
        True

    Schouten-Nijenhuis bracket for \(p=2\) and \(q=2\):

    Sage

        sage: d = M.multivector_field(2, name='d')
        sage: d[0,1], d[0,2], d[1,2] = x-y^2, x+z, z-x-1
        sage: s = c.bracket(d); s
        3-vector field [c,d] on the 3-dimensional differentiable manifold M
        sage: s.display()
        [c,d] = (-y^3 + (3*x + 1)*y - y^2 - x - z + 2) ∂/∂x∧∂/∂y∧∂/∂z

    Python

        >>> from sage.all import *
        >>> d = M.multivector_field(Integer(2), name='d')
        >>> d[Integer(0),Integer(1)], d[Integer(0),Integer(2)], d[Integer(1),Integer(2)] = x-y**Integer(2), x+z, z-x-Integer(1)
        >>> s = c.bracket(d); s
        3-vector field [c,d] on the 3-dimensional differentiable manifold M
        >>> s.display()
        [c,d] = (-y^3 + (3*x + 1)*y - y^2 - x - z + 2) ∂/∂x∧∂/∂y∧∂/∂z

    Let us check the component formula (with respect to the manifold’s default coordinate chart, i.e. `X`) for \(p=q=2\), taking into account the tensor antisymmetries:

    Sage

        sage: s[0,1,2] == - sum(c[i,0]*d[1,2].diff(i)
        ....:                 + c[i,1]*d[2,0].diff(i) + c[i,2]*d[0,1].diff(i)
        ....:                 + d[i,0]*c[1,2].diff(i) + d[i,1]*c[2,0].diff(i)
        ....:                 + d[i,2]*c[0,1].diff(i) for i in M.irange())
        True

    [Python]

        >>> from sage.all import *
        >>> s[Integer(0),Integer(1),Integer(2)] == - sum(c[i,Integer(0)]*d[Integer(1),Integer(2)].diff(i)
        ...                 + c[i,Integer(1)]*d[Integer(2),Integer(0)].diff(i) + c[i,Integer(2)]*d[Integer(0),Integer(1)].diff(i)
        ...                 + d[i,Integer(0)]*c[Integer(1),Integer(2)].diff(i) + d[i,Integer(1)]*c[Integer(2),Integer(0)].diff(i)
        ...                 + d[i,Integer(2)]*c[Integer(0),Integer(1)].diff(i) for i in M.irange())
        True

    Schouten-Nijenhuis bracket for \(p=1\) and \(q=3\):

    Sage

        sage: e = M.multivector_field(3, name='e')
        sage: e[0,1,2] = x+y*z+1
        sage: s = a.bracket(e); s
        3-vector field [a,e] on the 3-dimensional differentiable manifold M
        sage: s.display()
        [a,e] = (-(2*x + 1)*y + y^2 - (y^2 - x - 1)*z - z^2
         - 2*x - 2) ∂/∂x∧∂/∂y∧∂/∂z

    Python

        >>> from sage.all import *
        >>> e = M.multivector_field(Integer(3), name='e')
        >>> e[Integer(0),Integer(1),Integer(2)] = x+y*z+Integer(1)
        >>> s = a.bracket(e); s
        3-vector field [a,e] on the 3-dimensional differentiable manifold M
        >>> s.display()
        [a,e] = (-(2*x + 1)*y + y^2 - (y^2 - x - 1)*z - z^2
         - 2*x - 2) ∂/∂x∧∂/∂y∧∂/∂z

    Again, since \(p=1\), the bracket coincides with the Lie derivative:

    Sage

        sage: s == e.lie_derivative(a)
        True

    [Python]

        >>> from sage.all import *
        >>> s == e.lie_derivative(a)
        True

    Schouten-Nijenhuis bracket for \(p=2\) and \(q=3\):

    Sage

        sage: s = c.bracket(e); s
        4-vector field [c,e] on the 3-dimensional differentiable manifold M

    Python

        >>> from sage.all import *
        >>> s = c.bracket(e); s
        4-vector field [c,e] on the 3-dimensional differentiable manifold M

    Since on a 3-dimensional manifold, any 4-vector field is zero, we have:

    Sage

        sage: s.display()
        [c,e] = 0

    [Python]

        >>> from sage.all import *
        >>> s.display()
        [c,e] = 0

    Let us check the graded commutation law \([a,b] = -(-1)^{(p-1)(q-1)} [b,a]\) for various values of \(p\) and \(q\):

    Sage

        sage: f.bracket(a) == - a.bracket(f)  # p=0 and q=1
        True
        sage: f.bracket(c) == c.bracket(f)    # p=0 and q=2
        True
        sage: a.bracket(b) == - b.bracket(a)  # p=1 and q=1
        True
        sage: a.bracket(c) == - c.bracket(a)  # p=1 and q=2
        True
        sage: c.bracket(d) == d.bracket(c)    # p=2 and q=2
        True

    Python

        >>> from sage.all import *
        >>> f.bracket(a) == - a.bracket(f)  # p=0 and q=1
        True
        >>> f.bracket(c) == c.bracket(f)    # p=0 and q=2
        True
        >>> a.bracket(b) == - b.bracket(a)  # p=1 and q=1
        True
        >>> a.bracket(c) == - c.bracket(a)  # p=1 and q=2
        True
        >>> c.bracket(d) == d.bracket(c)    # p=2 and q=2
        True

    Let us check the graded Leibniz rule for \(p=1\) and \(q=1\):

    Sage

        sage: a.bracket(b.wedge(c)) == a.bracket(b).wedge(c) + b.wedge(a.bracket(c))  # long time
        True

    [Python]

        >>> from sage.all import *
        >>> a.bracket(b.wedge(c)) == a.bracket(b).wedge(c) + b.wedge(a.bracket(c))  # long time
        True

    as well as for \(p=2\) and \(q=1\):

    Sage

        sage: c.bracket(a.wedge(b)) == c.bracket(a).wedge(b) - a.wedge(c.bracket(b))  # long time
        True

    Python

        >>> from sage.all import *
        >>> c.bracket(a.wedge(b)) == c.bracket(a).wedge(b) - a.wedge(c.bracket(b))  # long time
        True

    Finally let us check the graded Jacobi identity for \(p=1\), \(q=1\) and \(r=2\):

    Sage

        sage: # long time
        sage: a_bc = a.bracket(b.bracket(c))
        sage: b_ca = b.bracket(c.bracket(a))
        sage: c_ab = c.bracket(a.bracket(b))
        sage: a_bc + b_ca + c_ab == 0
        True

    [Python]

        >>> from sage.all import *
        >>> # long time
        >>> a_bc = a.bracket(b.bracket(c))
        >>> b_ca = b.bracket(c.bracket(a))
        >>> c_ab = c.bracket(a.bracket(b))
        >>> a_bc + b_ca + c_ab == Integer(0)
        True

    as well as for \(p=1\), \(q=2\) and \(r=2\):

    Sage

        sage: # long time
        sage: a_cd = a.bracket(c.bracket(d))
        sage: c_da = c.bracket(d.bracket(a))
        sage: d_ac = d.bracket(a.bracket(c))
        sage: a_cd + c_da - d_ac == 0
        True

    Python

        >>> from sage.all import *
        >>> # long time
        >>> a_cd = a.bracket(c.bracket(d))
        >>> c_da = c.bracket(d.bracket(a))
        >>> d_ac = d.bracket(a.bracket(c))
        >>> a_cd + c_da - d_ac == Integer(0)
        True

  interior_product(form)

    Interior product with a differential form.

    If `self` is a multivector field \(A\) of degree \(p\) and \(B\) is a differential form of degree \(q\geq p\) on the same manifold as \(A\), the interior product of \(A\) by \(B\) is the differential form \(\iota_A B\) of degree \(q-p\) defined by

    \[(\iota_A B)_{i_1\ldots i_{q-p}} = A^{k_1\ldots k_p} B_{k_1\ldots k_p i_1\ldots i_{q-p}}\]

    Note

    `A.interior_product(B)` yields the same result as `A.contract(0,..., p-1, B, 0,..., p-1)` (cf. `contract()`), but `interior_product` is more efficient, the alternating character of \(A\) being not used to reduce the computation in `contract()`

    INPUT:

    - `form` – differential form \(B\) (instance of `DiffFormParal`); the degree of \(B\) must be at least equal to the degree of `self`

    OUTPUT:

    - scalar field (case \(p=q\)) or `DiffFormParal` (case \(p<q\)) representing the interior product \(\iota_A B\), where \(A\) is `self`

    See also

    `interior_product()` for the interior product of a differential form with a multivector field

    EXAMPLES:

    Interior product with \(p=1\) and \(q=1\) on 4-dimensional manifold:

    Sage

        sage: M = Manifold(4, 'M')
        sage: X.<t,x,y,z> = M.chart()
        sage: a = M.vector_field([x, 1+t^2, x*z, y-3], name='a')
        sage: b = M.one_form([-z^2, 2, x, x-y], name='b')
        sage: s = a.interior_product(b); s
        Scalar field i_a b on the 4-dimensional differentiable manifold M
        sage: s.display()
        i_a b: M → ℝ
           (t, x, y, z) ↦ x^2*z - x*z^2 + 2*t^2 + (x + 3)*y - y^2
            - 3*x + 2

    Python

        >>> from sage.all import *
        >>> M = Manifold(Integer(4), 'M')
        >>> X = M.chart(names=('t', 'x', 'y', 'z',)); (t, x, y, z,) = X._first_ngens(4)
        >>> a = M.vector_field([x, Integer(1)+t**Integer(2), x*z, y-Integer(3)], name='a')
        >>> b = M.one_form([-z**Integer(2), Integer(2), x, x-y], name='b')
        >>> s = a.interior_product(b); s
        Scalar field i_a b on the 4-dimensional differentiable manifold M
        >>> s.display()
        i_a b: M → ℝ
           (t, x, y, z) ↦ x^2*z - x*z^2 + 2*t^2 + (x + 3)*y - y^2
            - 3*x + 2

    In this case, we have \(\iota_a b = a^i b_i = a(b) = b(a)\):

    Sage

        sage: all([s == a.contract(b), s == a(b), s == b(a)])
        True

    [Python]

        >>> from sage.all import *
        >>> all([s == a.contract(b), s == a(b), s == b(a)])
        True

    Case \(p=1\) and \(q=3\):

    Sage

        sage: c = M.diff_form(3, name='c')
        sage: c[0,1,2], c[0,1,3] = x*y - z, -3*t
        sage: c[0,2,3], c[1,2,3] = t+x, y
        sage: s = a.interior_product(c); s
        2-form i_a c on the 4-dimensional differentiable manifold M
        sage: s.display()
        i_a c = (x^2*y*z - x*z^2 - 3*t*y + 9*t) dt∧dx
         + (-(t^2*x - t)*y + (t^2 + 1)*z - 3*t - 3*x) dt∧dy
         + (3*t^3 - (t*x + x^2)*z + 3*t) dt∧dz
         + ((x^2 - 3)*y + y^2 - x*z) dx∧dy
         + (-x*y*z - 3*t*x) dx∧dz + (t*x + x^2 + (t^2 + 1)*y) dy∧dz
        sage: s == a.contract(c)
        True

    Python

        >>> from sage.all import *
        >>> c = M.diff_form(Integer(3), name='c')
        >>> c[Integer(0),Integer(1),Integer(2)], c[Integer(0),Integer(1),Integer(3)] = x*y - z, -Integer(3)*t
        >>> c[Integer(0),Integer(2),Integer(3)], c[Integer(1),Integer(2),Integer(3)] = t+x, y
        >>> s = a.interior_product(c); s
        2-form i_a c on the 4-dimensional differentiable manifold M
        >>> s.display()
        i_a c = (x^2*y*z - x*z^2 - 3*t*y + 9*t) dt∧dx
         + (-(t^2*x - t)*y + (t^2 + 1)*z - 3*t - 3*x) dt∧dy
         + (3*t^3 - (t*x + x^2)*z + 3*t) dt∧dz
         + ((x^2 - 3)*y + y^2 - x*z) dx∧dy
         + (-x*y*z - 3*t*x) dx∧dz + (t*x + x^2 + (t^2 + 1)*y) dy∧dz
        >>> s == a.contract(c)
        True

    Case \(p=2\) and \(q=3\):

    Sage

        sage: d = M.multivector_field(2, name='d')
        sage: d[0,1], d[0,2], d[0,3] = t-x, 2*z, y-1
        sage: d[1,2], d[1,3], d[2,3] = z, y+t, 4
        sage: s = d.interior_product(c); s
        1-form i_d c on the 4-dimensional differentiable manifold M
        sage: s.display()
        i_d c = (2*x*y*z - 6*t^2 - 6*t*y - 2*z^2 + 8*t + 8*x) dt
         + (-4*x*y*z + 2*(3*t + 4)*y + 4*z^2 - 6*t) dx
         + (2*((t - 1)*x - x^2 - 2*t)*y - 2*y^2 - 2*(t - x)*z + 2*t
         + 2*x) dy + (-6*t^2 + 6*t*x + 2*(2*t + 2*x + y)*z) dz
        sage: s == d.contract(0, 1, c, 0, 1)
        True

    [Python]

        >>> from sage.all import *
        >>> d = M.multivector_field(Integer(2), name='d')
        >>> d[Integer(0),Integer(1)], d[Integer(0),Integer(2)], d[Integer(0),Integer(3)] = t-x, Integer(2)*z, y-Integer(1)
        >>> d[Integer(1),Integer(2)], d[Integer(1),Integer(3)], d[Integer(2),Integer(3)] = z, y+t, Integer(4)
        >>> s = d.interior_product(c); s
        1-form i_d c on the 4-dimensional differentiable manifold M
        >>> s.display()
        i_d c = (2*x*y*z - 6*t^2 - 6*t*y - 2*z^2 + 8*t + 8*x) dt
         + (-4*x*y*z + 2*(3*t + 4)*y + 4*z^2 - 6*t) dx
         + (2*((t - 1)*x - x^2 - 2*t)*y - 2*y^2 - 2*(t - x)*z + 2*t
         + 2*x) dy + (-6*t^2 + 6*t*x + 2*(2*t + 2*x + y)*z) dz
        >>> s == d.contract(Integer(0), Integer(1), c, Integer(0), Integer(1))
        True

  wedge(other)

    Exterior product of `self` with another multivector field.

    INPUT:

    - `other` – another multivector field

    OUTPUT:

    - instance of `MultivectorFieldParal` representing the exterior product `self ∧ other`

    EXAMPLES:

    Exterior product of a vector field and a 2-vector field on a 3-dimensional manifold:

    Sage

        sage: M = Manifold(3, 'M', start_index=1)
        sage: X.<x,y,z> = M.chart()
        sage: a = M.vector_field([2, 1+x, y*z], name='a')
        sage: b = M.multivector_field(2, name='b')
        sage: b[1,2], b[1,3], b[2,3] = y^2, z+x, z^2
        sage: a.display()
        a = 2 ∂/∂x + (x + 1) ∂/∂y + y*z ∂/∂z
        sage: b.display()
        b = y^2 ∂/∂x∧∂/∂y + (x + z) ∂/∂x∧∂/∂z + z^2 ∂/∂y∧∂/∂z
        sage: s = a.wedge(b); s
        3-vector field a∧b on the 3-dimensional differentiable manifold M
        sage: s.display()
        a∧b = (-x^2 + (y^3 - x - 1)*z + 2*z^2 - x) ∂/∂x∧∂/∂y∧∂/∂z

    Python

        >>> from sage.all import *
        >>> M = Manifold(Integer(3), 'M', start_index=Integer(1))
        >>> X = M.chart(names=('x', 'y', 'z',)); (x, y, z,) = X._first_ngens(3)
        >>> a = M.vector_field([Integer(2), Integer(1)+x, y*z], name='a')
        >>> b = M.multivector_field(Integer(2), name='b')
        >>> b[Integer(1),Integer(2)], b[Integer(1),Integer(3)], b[Integer(2),Integer(3)] = y**Integer(2), z+x, z**Integer(2)
        >>> a.display()
        a = 2 ∂/∂x + (x + 1) ∂/∂y + y*z ∂/∂z
        >>> b.display()
        b = y^2 ∂/∂x∧∂/∂y + (x + z) ∂/∂x∧∂/∂z + z^2 ∂/∂y∧∂/∂z
        >>> s = a.wedge(b); s
        3-vector field a∧b on the 3-dimensional differentiable manifold M
        >>> s.display()
        a∧b = (-x^2 + (y^3 - x - 1)*z + 2*z^2 - x) ∂/∂x∧∂/∂y∧∂/∂z

    Check:

    Sage

        sage: s[1,2,3] == a[1]*b[2,3] + a[2]*b[3,1] + a[3]*b[1,2]
        True

    [Python]

        >>> from sage.all import *
        >>> s[Integer(1),Integer(2),Integer(3)] == a[Integer(1)]*b[Integer(2),Integer(3)] + a[Integer(2)]*b[Integer(3),Integer(1)] + a[Integer(3)]*b[Integer(1),Integer(2)]
        True

    Exterior product with a scalar field:

    Sage

        sage: f = M.scalar_field(x, name='f')
        sage: s = b.wedge(f); s
        2-vector field f*b on the 3-dimensional differentiable manifold M
        sage: s.display()
        f*b = x*y^2 ∂/∂x∧∂/∂y + (x^2 + x*z) ∂/∂x∧∂/∂z + x*z^2 ∂/∂y∧∂/∂z
        sage: s == f*b
        True
        sage: s == f.wedge(b)
        True

    Python

        >>> from sage.all import *
        >>> f = M.scalar_field(x, name='f')
        >>> s = b.wedge(f); s
        2-vector field f*b on the 3-dimensional differentiable manifold M
        >>> s.display()
        f*b = x*y^2 ∂/∂x∧∂/∂y + (x^2 + x*z) ∂/∂x∧∂/∂z + x*z^2 ∂/∂y∧∂/∂z
        >>> s == f*b
        True
        >>> s == f.wedge(b)
        True