SAGE tip: More about Differential Forms

In a previous post we discuss the definition of the coordinated patch on a manifold, how to define differential forms, wedge them or calculate their exterior derivative… even simplify’em.

This time a zero form will be defined and a list of forms will be created… So, let’s begin!

Define a 0-form

Once created the coordinated patch and the differential forms algebra

sage: reset()
sage: var('t,x,y,z')
sage: U = CoordinatePatch((t,x,y,z))
sage: Omega = DifferentialForms(U)

A 0-form is defined as an element of , but the value of the 0-form is given inside the declaration command,

sage: A = DifferentialForm(Omega, 0, exp(x*y))

I tried addition, multiplication, wedge product and exterior differentiation on 0-forms and they worked!

Of course you can combine them with forms of different degrees.

New method of defining a form

I wrote to Joris this morning… but before he was able to answer, from the documentation of the differential form package.

When one calls the generators of the differential form,

sage: Omega.gen(1)
dx

and the result is a differential form… Thus, one can assign a form as follow,

sage: A = sin(x)* Omega.gen(2)
sage: B = cos(y) * Omega.gen(0)
sage: C = sin(z) *Omega.gen(1)
sage: D = cos(y) * Omega.gen(2)

And this forms can be wedged, differentiated, et cetera. 🙂

List of forms

Finally, after discovering the above behavior I tried the following, a list of differential forms ;-), for example,

sage: pro = matrix([[A, B], [C, D]])
sage: for i in range(2):
...       for j in range(2):
...           show(pro[i,j].diff())

returns

This implies that somehow one can manage a series of forms by using list properties… I expect to go deeper on this subject in the future! 😀

Enjoy people!!!

Dox

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SAGE tip: GRmodule. Day 07.

This code is supposed to be (if some one does the work in the future) located in sage.tensor.differential_form_element.

The code presented below is a slight modification of Joris code for differential forms manipulation on SAGE.

Needed modules

from sage.symbolic.ring import SymbolicRing, SR
from sage.rings.ring_element import RingElement
from sage.algebras.algebra_element import AlgebraElement
from sage.rings.integer import Integer
from sage.combinat.permutation import Permutation

The advantage of using this is that, tensors defined here are an Algebra element, not just a python object as in the previous code.

The sage.combinat.permutation won’t be used (yet), but could be useful if tensor symmetries are defined.

TensorFormatter

class TensorFormatter:
    r"""
    This class contains all the functionality to print a tensor in a
    graphically pleasing way.  This class is called by the ``_latex_`` and
    ``_repr_`` methods of the Tensor class.
    """
    def __init__(self, space):
        r"""
        Construct a tensor formatter.  See
        ``TensorFormatter`` for more information.
        """
        self._space = space

    def repr(self, comp, fun):
        r"""
        String representation of a primitive tensor, i.e. a function
        times a tensor product of d's of the coordinate functions.

        INPUT:

        - ``comp`` -- a subscript of a differential form.

        - ``fun`` -- the component function of this form.

        EXAMPLES::

            sage: from sage.tensor.tensor_element import TensorFormatter
            sage: x, y, z = var('x, y, z')
            sage: U = CoordinatePatch((x, y, z))
            sage: D = TensorFormatter(U)
            sage: D.repr((0, 1), z^3)
            'z^3*dx@dy'

        """

        str = "@".join( \
            [('d%s' % self._space.coordinate(c).__repr__()) for c in comp])

        if fun == 1 and len(comp) > 0:
            # We have a non-trivial form whose component function is 1,
            # so we just return the formatted form part and ignore the 1.
            return str
        else:
            funstr = fun._repr_()

            if not self._is_atomic(funstr):
                funstr = '(' + funstr + ')'

            if len(str) > 0:
                return funstr + "*" + str
            else:
                return funstr

    def latex(self, comp, fun):
        r"""
        Latex representation of a primitive differential form, i.e. a function
        times a tensor product of d's of the coordinate functions.

        INPUT:

        - ``comp`` -- a subscript of a differential form.

        - ``fun`` -- the component function of this form.

        EXAMPLES::

            sage: from sage.tensor.tensor_element import TensorFormatter
            sage: x, y, z = var('x, y, z')
            sage: U = CoordinatePatch((x, y, z))
            sage: D = TensorFormatter(U)
            sage: D.latex((0, 1), z^3)
            'z^{3} d x \otimes d y'

        """

        from sage.misc.latex import latex

        str = " \otimes ".join( \
            [('d %s' % latex(self._space.coordinate(c))) for c in comp])

        if fun == 1 and len(comp) > 0:
            return str
        else:
            funstr = latex(fun)

            if not self._is_atomic(funstr):
                funstr = '(' + funstr + ')'

            return funstr + " " + str

    def _is_atomic(self, str):
        r"""
        Helper function to check whether a given string expression
        is atomic.

        EXAMPLES::

            sage: x, y, z = var('x, y, z')
            sage: U = CoordinatePatch((x, y, z))
            sage: from sage.tensor.tensor_element import TensorFormatter
            sage: D = TensorFormatter(U)
            sage: D._is_atomic('a + b')
            False
            sage: D._is_atomic('(a + b)')
            True
        """
        level = 0
        for n, c in enumerate(str):
            if c == '(':
                level += 1
            elif c == ')':
                level -= 1

            if c == '+' or c == '-':
                if level == 0 and n > 0:
                    return False
        return True

The only I’ve changed here is “DifferentialForm” by “Tensor” and “\wedge” by “\otimes”

The above code allows to write the tensor product in a basis, . The chosen symbol for denoting the tensor product was @.

Tensor Class

This code is incomplete due to:

I’ve not defined the TensorsAlgebra, which should be done in parallel.

There are a lot of attributes not presented in this class.

class Tensor(AlgebraElement):
    r"""
    Tensor class.
    """

    def __init__(self, parent, degree, fun = None):
        r"""
        Construct a tensor.

        INPUT:

        - ``parent`` -- Parent algebra of tensors.

        - ``degree`` -- Degree of the tensor.

        - ``fun`` (default: None) -- Initialize this differential form with the given function.  If the degree is not zero, this argument is silently ignored.

        EXAMPLES::

            sage: x, y, z = var('x, y, z')
            sage: F = Tensors(); F
            Algebra of tensors in the variables x, y, z
            sage: f = Tensor(F, 0, sin(z)); f
            sin(z)

        """

        from sage.tensor.tensorss import Tensors
        if not isinstance(parent, Tensors):
            raise TypeError, "Parent not an algebra of tensors."

        RingElement.__init__(self, parent)

        self._degree = degree
        self._components = {}

        if degree == 0 and fun is not None:
            self.__setitem__([], fun)

    def __getitem__(self, subscript):
        r"""
        Return a given component of the tensor.

        INPUT:

        - ``subscript``: subscript of the component.  Must be an integer
        or a list of integers.

        EXAMPLES::

            sage: x, y, z = var('x, y, z')
            sage: F = Tensors(); F
            Algebra of tensors in the variables x, y, z
            sage: f = Tensor(F, 0, sin(x*y)); f
            sin(x*y)
            sage: f[()]
            sin(x*y)
        """

        if isinstance(subscript, (Integer, int)):
            subscript = (subscript, )
        else:
            subscript = tuple(subscript)

        dim = self.parent().base_space().dim()
        if any([s >= dim for s in subscript]):
            raise ValueError, "Index out of bounds."

        if len(subscript) != self._degree:
            raise TypeError, "%s is not a subscript of degree %s" %\
                (subscript, self._degree)

        """sign, subscript = sort_subscript(subscript)"""

        if subscript in self._components:
            return sign*self._components[subscript]
        else:
            return 0

    def __setitem__(self, subscript, fun):
        r"""
        Modify a given component of the tensor.

        INPUT:

        - ``subscript``: subscript of the component.  Must be an integer or a list of integers.

        EXAMPLES::

            sage: F = Tensors(); F
            Algebra of tensors in the variables x, y, z
            sage: f = Tensor(F, 2)
            sage: f[1, 2] = x; f
            x*dy@dz
        """

        if isinstance(subscript, (Integer, int)):
            subscript = (subscript, )
        else:
            subscript = tuple(subscript)

        dim = self.parent().base_space().dim()
        if any([s >= dim for s in subscript]):
            raise ValueError, "Index out of bounds."

        if len(subscript) != self._degree:
            raise TypeError, "%s is not a subscript of degree %s" %\
                (subscript, self._degree)

        """sign, subscript = sort_subscript(subscript)"""
        self._components[subscript] = SR(fun)

Ok, so again I’ve changed “DifferentialForm(s)” by “Tensor(s)”, drop the permutation of indices (’cause tensors do not need to be neither symmetric nor anti-symmetric.

I’ll keep working with this code… It’s all by now. Oh! I’ll post next week the rest of the code based in Sergey’s GRPy.

Enjoy.

Dox

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SAGE tip: GRmodule. Day 06.

I’ve just updated the SAGE worksheet which uses the definitions described in the previous posts.

There are some explanations in text format

The code has been hidden… because is long.

Moreover… I’ve discover something really amazing! Joris Vankerschaver‘s code of the differential form package in SAGE. Thus, I could use some ideas from Joris’ code to improve GRmodule. Nice, Isn’t it?

Let’s hope I could so something nice this weekend!

Don’t forget check the worksheet, and post some comment for feedback! 😉

Enjoy!

Dox

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SAGE tip: GRmodule. Day 05

Hi everyone!

This time the Christoffel connection will be defined.

The code

As usual, here is the code:

class Christoffel(Tensor):
    '''The class to represent Christoffel Symbols of the second kind. Please
        note that while it inherits from Tensor, Christoffel symbols are
        NOT tensors'''

    def __init__(self,metr,symbol='C',rank=(1,2),sh=(1,-1,-1)):

        # The metric
        self.g_down = metr

        # Since we have a metric we do indeed have a coordinate system
        self.rep  = self.g_down.rep

        self.g_up = metr.inverse

        # Please note that this call will trigger a call to allocate in
        # the Tensor class, but the allocate will actually be the allocate
        # defined below
        super(Christoffel,self).__init__(symbol,rank,sh,coords=metr.coord)

    def allocate(self,rank):
        Tensor.allocate(self,rank)
        # Now that we have allocated things, time to actually compute things
        for i in xrange(self.dim):
            for k in xrange(self.dim):
                for l in xrange(self.dim):
                    sum = 0
                    for m in xrange(self.dim):
                        term1 = diff(self.g_down[m,k],self.g_down.coords[l])
                        term2 = diff(self.g_down[m,l],self.g_down.coords[k])
                        term3 = diff(self.g_down[k,l],self.g_down.coords[m])

                        tr = self.g_up[i,m] * (term1+term2-term3)

                        sum += tr
                    res = sum/2
                    self.components[i,k,l] = res
        self.getNonZero()

This code is almost a copy of Sergey’s one, except for the use of xrange instead of np.arange, and the fact that I’ve dropped the minus signs denoting the shape of the tensors.

Sage implementation

This time I won’t present a Python file, but a SAGE file, GRmodule.

Enjoy!

Dox

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SAGE tip: Working with Differential Forms.

A few weeks ago I realize there is a package for working with differential forms in SAGE. So, I’ve tried to take advantage of that!

This is what I’ve learnt so far.

Define your manifold

If you work with differential forms you should know (probably better than me), that you define forms in a coordinate patch… rarely globally.

I’d like to start with something like this!

sage: reset()
sage: # The variables
sage: var('theta,phi,r')
sage: var("bps", latex_name=r"\bar{\psi}")
sage: var("bph", latex_name=r"\bar{\phi}")
sage: var("bth", latex_name=r"\bar{\theta}")
sage: # The coordinate system
sage: coords = [r,theta, phi, bps, bth, bph]
sage: U = CoordinatePatch((r,theta, phi, bps, bth, bph))
sage: Omega = DifferentialForms(U)

From a previous post a weird declaration of variables has been made.

But don’t lose yourself into the code!!!… the important here is:

Seven variables have been declared,

These are converted into a list,

and also are the coordinated of the Coordinate Patch.

Finally, is the algebra of differential forms. Just like , where is the manifold (in this case the patch).

Define the differential forms

Differential form have to be defined… like this,

sage: A = DifferentialForm(Omega, 1)
sage: A[1] = -1/4
sage: A[2] = 1/4*(-cos(theta) - sin(theta))
sage: A[3] = cos(bps)*sin(bth)*sin(bph)
sage: A[4] = sin(bps)*cos(bth)*sin(bph)
sage: A[5] = sin(bps)*sin(bth)*cos(bph)

The first line defines as an element of .

The rest of the lines are for setting the components of . Remember that the coordinates run from 0 to 6 in this example! 😛 And, of course, they are ordered strictly as we declared them, i.e.,

Exterior Differentiation

Once the form has been declared, one might differentiate it by using the diff command,

sage: A.diff()

returns

Showing the forms

It’s very useful to see the formulas in a nice written way… this is one of the features I love the more from SAGE, to see your form, use

 sage: show(A)

Wedge Product

Of course one of the most important operations when working with forms is the wedge product. For using this, try

sage: C = A.wedge(A.diff())
sage: show(C)

it vanishes. One might try to define another form,

sage: B = DifferentialForm(Omega,2)
sage: B[0,1] = sin(bth)

Note that , and has only one non-vanishing component.

Thus, is calculated,

sage: A.wedge(B)

One can also define multiple wedge product,

sage: A.wedge(B.diff()).wedge(B)

Needless to say one can multiply a form by a function, or number.

Simplifying a form

Forms have not implemented the simplify_full attribute, but their components, which are functions, do. So, after a very complicated calculation one might try to implement a long routine of simplification, say,

sage: D = A.wedge(B.diff()).wedge(B)
for i in xrange(Omega.ngens()):
    for j in xrange(i+1, Omega.ngens()):
        for k in xrange(j+1, Omega.ngens()):
            for l in xrange(k+1, Omega.ngens()):
                for m in xrange(l+1, Omega.ngens()):
                    D[i,j,k,l,m] = D[i,j,k,l,m].simplify_full()

In the above, Omega.ngens() returns the dimension of the patch. Additionally, each index runs from the value of the previous plus one, in order to avoid repetition (due to anti-symmetric property) or the zeros values.

if someone find a better way, doesn’t hesitate in post a comment!!! 🙂

Well, I think this is much of it! I’ll keep you posted, in case I learn more about ot.

Enjoy life! and Happy Valentine’s day! 😉

Dox

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