Revision history [back]

At the moment (SageMath 8.7), differential forms of degree $p$ on a manifold $M$ are considered to belong to (i.e. their parent is) the set $\Omega^p(M)$ of all differential forms of degree $p$, which is a $C^\infty(M)$-module and not a ring. Hence one cannot construct matrices from them. However, since SageMath 8.8.beta3 (see the ticket #27584 for details), it is possible to consider the graded algebra $$\Omega^(M) = \bigoplus^n_{p=0} \Omega^p(M), $$ where $n = \mathrm{dim}\ M$. $\Omega^(M)$ is a ring, the multiplication of which is the wedge product. It is therefore possible to form matrices from its elements. The latter are called mixed differential forms, so that, continuing from your example, one may write

sage: aa = M.mixed_form(comp=[0, 0, a], name='aa')
sage: aa.parent()
Graded algebra Omega^*(M) of mixed differential forms on the 2-dimensional complex manifold M
sage: aa.display(eU)
aa = [0] + [0] + [(x*y^2 + 2*x) dx/\dy]

It is now possible to form a matrix and use the standard matrix operations:

sage: A = matrix([[aa, aa], [aa, aa]])
sage: A
[Mixed differential form aa on the 2-dimensional complex manifold M 
 Mixed differential form aa on the 2-dimensional complex manifold M]
[Mixed differential form aa on the 2-dimensional complex manifold M 
 Mixed differential form aa on the 2-dimensional complex manifold M]
sage: A[0,1]
Mixed differential form aa on the 2-dimensional complex manifold M
sage: det(A)
Mixed differential form aa/\aa-aa/\aa on the 2-dimensional complex manifold M
sage: det(A).display(eU)
aa/\aa-aa/\aa = [0] + [0] + [0]

The above feature will be available in the next stable release of SageMath (8.8), which should appear in a few weeks. Meanwhile you can install the latest development version (8.8.beta6 as of today) to use it, see e.g. the section "1. Starting from the latest sources" in this page.

$$\Omega^*(M) = \bigoplus^n_{p=0} \Omega^p(M), $$ $$

where $n = \mathrm{dim}\ M$. $\Omega^(M)$ is a ring, the multiplication of which is the wedge product. It is therefore possible to form matrices from its elements. The latter are called ~~mixed~~ *mixed differential forms, so that, continuing from your example, one may write

sage: aa = M.mixed_form(comp=[0, 0, a], name='aa') sage: aa.parent() Graded algebra Omega^*(M) of mixed differential forms on the 2-dimensional complex manifold M sage: aa.display(eU) aa = [0] + [0] + [(x*y^2 + 2*x) dx/\dy]

It is now possible to form a matrix and use the standard matrix operations:

sage: A = matrix([[aa, aa], [aa, aa]]) sage: A [Mixed differential form aa on the 2-dimensional complex manifold M Mixed differential form aa on the 2-dimensional complex manifold M] [Mixed differential form aa on the 2-dimensional complex manifold M Mixed differential form aa on the 2-dimensional complex manifold M] sage: A[0,1] Mixed differential form aa on the 2-dimensional complex manifold M sage: det(A) Mixed differential form aa/\aa-aa/\aa on the 2-dimensional complex manifold M sage: det(A).display(eU) aa/\aa-aa/\aa = [0] + [0] + [0]

The above feature will be available in the next stable release of SageMath (8.8), which should appear in a few weeks. Meanwhile you can install the latest development version (8.8.beta6 as of today) to use it, see e.g. the section "1. Starting from the latest sources" in this page.
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3
No.3 Revision

updated 2019-05-27 14:14:29 +0200

At the moment (SageMath 8.7), differential forms of degree $p$ on a manifold $M$ are considered to belong to (i.e. their parent is) the set $\Omega^p(M)$ of all differential forms of degree $p$, which is a $C^\infty(M)$-module and not a ring. Hence one cannot construct matrices from them. However, since SageMath 8.8.beta3 (see the ticket #27584 for details), it is possible to consider the graded algebra

$$\Omega^*(M) = \bigoplus^n_{p=0} \Omega^p(M), $$

where $n = \mathrm{dim}\ M$. The set $\Omega^(M)$ is a ring, the multiplication of which is the wedge product. It is therefore possible to form matrices from its elements. The latter are called *mixed differential forms, so that, continuing from your example, one may write

sage: aa = M.mixed_form(comp=[0, 0, a], name='aa') sage: aa.parent() Graded algebra Omega^*(M) of mixed differential forms on the 2-dimensional complex manifold M sage: aa.display(eU) aa = [0] + [0] + [(x*y^2 + 2*x) dx/\dy]

It is now possible to form a matrix and use the standard matrix operations:

sage: A = matrix([[aa, aa], [aa, aa]]) sage: A [Mixed differential form aa on the 2-dimensional complex manifold M Mixed differential form aa on the 2-dimensional complex manifold M] [Mixed differential form aa on the 2-dimensional complex manifold M Mixed differential form aa on the 2-dimensional complex manifold M] sage: A[0,1] Mixed differential form aa on the 2-dimensional complex manifold M sage: det(A) Mixed differential form aa/\aa-aa/\aa on the 2-dimensional complex manifold M sage: det(A).display(eU) aa/\aa-aa/\aa = [0] + [0] + [0]

The above feature will be available in the next stable release of SageMath (8.8), which should appear in a few weeks. Meanwhile you can install the latest development version (8.8.beta6 as of today) to use it, see e.g. the section "1. Starting from the latest sources" in this page.

4
No.4 Revision

updated 2019-05-27 14:15:15 +0200

At the moment (SageMath 8.7), differential forms of degree $p$ on a manifold $M$ are considered to belong to (i.e. their parent is) the set $\Omega^p(M)$ of all differential forms of degree $p$, which is a $C^\infty(M)$-module and not a ring. Hence one cannot construct matrices from them. However, since SageMath 8.8.beta3 (see the ticket #27584 for details), it is possible to consider the graded algebra

$$\Omega^*(M) = \bigoplus^n_{p=0} \Omega^p(M), $$

where $n = \mathrm{dim}\ M$. The set ~~$\Omega^~~$\Omega^{~~(M)$~~ }(M)$ is a ring, the multiplication of which is the wedge product. It is therefore possible to form matrices from its elements. The latter are called *mixed differential forms, so that, continuing from your example, one may write

sage: aa = M.mixed_form(comp=[0, 0, a], name='aa') sage: aa.parent() Graded algebra Omega^*(M) of mixed differential forms on the 2-dimensional complex manifold M sage: aa.display(eU) aa = [0] + [0] + [(x*y^2 + 2*x) dx/\dy]

It is now possible to form a matrix and use the standard matrix operations:

sage: A = matrix([[aa, aa], [aa, aa]]) sage: A [Mixed differential form aa on the 2-dimensional complex manifold M Mixed differential form aa on the 2-dimensional complex manifold M] [Mixed differential form aa on the 2-dimensional complex manifold M Mixed differential form aa on the 2-dimensional complex manifold M] sage: A[0,1] Mixed differential form aa on the 2-dimensional complex manifold M sage: det(A) Mixed differential form aa/\aa-aa/\aa on the 2-dimensional complex manifold M sage: det(A).display(eU) aa/\aa-aa/\aa = [0] + [0] + [0]

The above feature will be available in the next stable release of SageMath (8.8), which should appear in a few weeks. Meanwhile you can install the latest development version (8.8.beta6 as of today) to use it, see e.g. the section "1. Starting from the latest sources" in this page.

5
No.5 Revision

updated 2019-05-27 14:15:39 +0200

At the moment (SageMath 8.7), differential forms of degree $p$ on a manifold $M$ are considered to belong to (i.e. their parent is) the set $\Omega^p(M)$ of all differential forms of degree $p$, which is a $C^\infty(M)$-module and not a ring. Hence one cannot construct matrices from them. However, since SageMath 8.8.beta3 (see the ticket #27584 for details), it is possible to consider the graded algebra

$$\Omega^*(M) = \bigoplus^n_{p=0} \Omega^p(M), $$

where $n = \mathrm{dim}\ M$. The set ~~$\Omega^{~~$ \Omega^{~~}(M)$~~ }(M) $ is a ring, the multiplication of which is the wedge product. It is therefore possible to form matrices from its elements. The latter are called *mixed differential forms, so that, continuing from your example, one may write

sage: aa = M.mixed_form(comp=[0, 0, a], name='aa') sage: aa.parent() Graded algebra Omega^*(M) of mixed differential forms on the 2-dimensional complex manifold M sage: aa.display(eU) aa = [0] + [0] + [(x*y^2 + 2*x) dx/\dy]

It is now possible to form a matrix and use the standard matrix operations:

sage: A = matrix([[aa, aa], [aa, aa]]) sage: A [Mixed differential form aa on the 2-dimensional complex manifold M Mixed differential form aa on the 2-dimensional complex manifold M] [Mixed differential form aa on the 2-dimensional complex manifold M Mixed differential form aa on the 2-dimensional complex manifold M] sage: A[0,1] Mixed differential form aa on the 2-dimensional complex manifold M sage: det(A) Mixed differential form aa/\aa-aa/\aa on the 2-dimensional complex manifold M sage: det(A).display(eU) aa/\aa-aa/\aa = [0] + [0] + [0]

The above feature will be available in the next stable release of SageMath (8.8), which should appear in a few weeks. Meanwhile you can install the latest development version (8.8.beta6 as of today) to use it, see e.g. the section "1. Starting from the latest sources" in this page.

6
No.6 Revision

updated 2019-05-27 14:16:10 +0200

At the moment (SageMath 8.7), differential forms of degree $p$ on a manifold $M$ are considered to belong to (i.e. their parent is) the set $\Omega^p(M)$ of all differential forms of degree $p$, which is a $C^\infty(M)$-module and not a ring. Hence one cannot construct matrices from them. However, since SageMath 8.8.beta3 (see the ticket #27584 for details), it is possible to consider the graded algebra

$$\Omega^*(M) = \bigoplus^n_{p=0} \Omega^p(M), $$

where $n = \mathrm{dim}\ M$. The set $ \Omega^{}(M) \Omega (M) $ is a ring, the multiplication of which is the wedge product. It is therefore possible to form matrices from its elements. The latter are called *mixed mixed differential forms, so that, continuing from your example, one may write

sage: aa = M.mixed_form(comp=[0, 0, a], name='aa') sage: aa.parent() Graded algebra Omega^*(M) of mixed differential forms on the 2-dimensional complex manifold M sage: aa.display(eU) aa = [0] + [0] + [(x*y^2 + 2*x) dx/\dy]

It is now possible to form a matrix and use the standard matrix operations:

sage: A = matrix([[aa, aa], [aa, aa]]) sage: A [Mixed differential form aa on the 2-dimensional complex manifold M Mixed differential form aa on the 2-dimensional complex manifold M] [Mixed differential form aa on the 2-dimensional complex manifold M Mixed differential form aa on the 2-dimensional complex manifold M] sage: A[0,1] Mixed differential form aa on the 2-dimensional complex manifold M sage: det(A) Mixed differential form aa/\aa-aa/\aa on the 2-dimensional complex manifold M sage: det(A).display(eU) aa/\aa-aa/\aa = [0] + [0] + [0]

The above feature will be available in the next stable release of SageMath (8.8), which should appear in a few weeks. Meanwhile you can install the latest development version (8.8.beta6 as of today) to use it, see e.g. the section "1. Starting from the latest sources" in this page.

7
No.7 Revision

updated 2019-05-27 14:16:50 +0200

At the moment (SageMath 8.7), differential forms of degree $p$ on a manifold $M$ are considered to belong to (i.e. their parent is) the set $\Omega^p(M)$ of all differential forms of degree $p$, which is a $C^\infty(M)$-module and not a ring. Hence one cannot construct matrices from them. However, since SageMath 8.8.beta3 (see the ticket #27584 for details), it is possible to consider the graded algebra

$$\Omega^*(M) = \bigoplus^n_{p=0} \Omega^p(M), $$

where $n = \mathrm{dim}\ M$. The set $ ~~\Omega~~ \Omega^* (M) $ is a ring, the multiplication of which is the wedge product. It is therefore possible to form matrices from its elements. The latter are called mixed differential forms, so that, continuing from your example, one may write

sage: aa = M.mixed_form(comp=[0, 0, a], name='aa') sage: aa.parent() Graded algebra Omega^*(M) of mixed differential forms on the 2-dimensional complex manifold M sage: aa.display(eU) aa = [0] + [0] + [(x*y^2 + 2*x) dx/\dy]

It is now possible to form a matrix and use the standard matrix operations:

sage: A = matrix([[aa, aa], [aa, aa]]) sage: A [Mixed differential form aa on the 2-dimensional complex manifold M Mixed differential form aa on the 2-dimensional complex manifold M] [Mixed differential form aa on the 2-dimensional complex manifold M Mixed differential form aa on the 2-dimensional complex manifold M] sage: A[0,1] Mixed differential form aa on the 2-dimensional complex manifold M sage: det(A) Mixed differential form aa/\aa-aa/\aa on the 2-dimensional complex manifold M sage: det(A).display(eU) aa/\aa-aa/\aa = [0] + [0] + [0]

The above feature will be available in the next stable release of SageMath (8.8), which should appear in a few weeks. Meanwhile you can install the latest development version (8.8.beta6 as of today) to use it, see e.g. the section "1. Starting from the latest sources" in this page.

8
No.8 Revision

updated 2019-05-27 14:23:43 +0200

At the moment (SageMath 8.7), differential forms of degree $p$ on a manifold $M$ are considered to belong to (i.e. their parent is) the set $\Omega^p(M)$ of all differential forms of degree $p$, which is a $C^\infty(M)$-module and not a ring. Hence one cannot construct matrices from them. However, since SageMath 8.8.beta3 (see the ticket #27584 for details), it is possible to consider the graded algebra

$$\Omega^*(M) = \bigoplus^n_{p=0} \Omega^p(M), $$

where $n = \mathrm{dim}\ M$. The set $ \Omega^* (M) $ is a ring, the multiplication of which is the wedge product. It is therefore possible to form matrices from its elements. The latter are called mixed differential forms, so that, continuing from your example, one may write

sage: aa = M.mixed_form(comp=[0, 0, a], name='aa') sage: aa.parent() Graded algebra Omega^*(M) of mixed differential forms on the 2-dimensional complex manifold M sage: aa.display(eU) aa = [0] + [0] + [(x*y^2 + 2*x) dx/\dy]

It is now possible to form a matrix and use the standard matrix operations:

sage: A = matrix([[aa, aa], [aa, aa]]) sage: A [Mixed differential form aa on the 2-dimensional complex manifold M Mixed differential form aa on the 2-dimensional complex manifold M] [Mixed differential form aa on the 2-dimensional complex manifold M Mixed differential form aa on the 2-dimensional complex manifold M] sage: A.base_ring() Graded algebra Omega^*(M) of mixed differential forms on the 2-dimensional complex manifold M sage: A[0,1] Mixed differential form aa on the 2-dimensional complex manifold M sage: det(A) Mixed differential form aa/\aa-aa/\aa on the 2-dimensional complex manifold M sage: det(A).display(eU) aa/\aa-aa/\aa = [0] + [0] + [0]

The above feature will be available in the next stable release of SageMath (8.8), which should appear in a few weeks. Meanwhile you can install the latest development version (8.8.beta6 as of today) to use it, see e.g. the section "1. Starting from the latest sources" in this page.