I vaguely remember the trajectory of a relativistic particle presented in
jackson's electrodynamics. I suppose it's convoluted enough for use in
tests.
A simpler one will be some surface density over a strange surface. For
example the total charge of a unitary surface charge distribution on a
spherical surface (of radius = 1) will be:
<delta(sqrt(x**2+y**2+z**2)-1) | 1> = 4 pi
Well this was just the surface of the sphere so it was not that
interesting, but one can calculate for example a field using such integral.
I'll try to dig some nice examples.
PS Mathematica can calculate those (at least in 1D and 2D)
In[6]:= Integrate[DiracDelta[-1 + Sqrt[x^2]], {x, -Infinity, Infinity}]
Out[6]= 2
In[7]:= Integrate[DiracDelta[-1 + Sqrt[x^2 + y^2]], {x, -Infinity,
Infinity}, {y, -Infinity, Infinity}]
Out[7]= 2 Pi
On 30 December 2011 01:22, Aaron Meurer <[email protected]> wrote:
> Does someone know of an example integral containing a delta function
> with a non-linear argument, with the correct answer? That would help
> with trying to find the algorithm.
>
> Perhaps it can be done by a change of variable, to make the argument
> linear. You probably have to be careful about some things, though,
> since delta functions don't follow all the "rules" that normal
> functions do.
>
> Aaron Meurer
>
> On Tue, Dec 20, 2011 at 12:51 PM, Matthew Rocklin <[email protected]>
> wrote:
> > @Tom - I agree that it makes sense to separate out the common easy case
> from
> > the very rare very difficult case.
> > @Stephan - In the multivariate case the zeros are likely to consist of
> > continuous lines/surfaces/manifolds of co-dimension 1. We'll need to do
> an
> > integral over complex surfaces. This could be challenging.
> >
> > From my experience deltaintegrate has some issues, even in the linear
> case.
> > The code also seems like it might be more complex than necessary,
> allowing
> > odd bugs to filter through. Wolfram handles deltas more robustly than
> SymPy
> > does so at least we know that there exists a nicer solution. I wonder if
> a
> > system something like what Stephan outlined would be cleaner/more robust.
> >
> >
> > On Tue, Dec 20, 2011 at 11:26 AM, [email protected]
> > <[email protected]> wrote:
> >>
> >> @Tom, the operation is well defined for all sufficiently nice functions.
> >> @Matthew, The notation that is used there is a bit unclear (for me). The
> >> domain is all zeros of g and the sigma is just some measure theory
> notation
> >> meaning that you should actually take the sum.
> >>
> >> To solve this in sympy you can try the following thing (it depends on
> how
> >> good 'solve' is in sympy):
> >>
> >> 0) to solve integrate(delta(f(x,y,...)*g(x,y,...)), (x,...), (y,...),
> ...)
> >> 1) take out the g(x,y,...) function from the delta function
> >> 2) find the zeros of g(...) (presumably using 'solve')
> >> 3) evaluate the derivatives of g(...) at those zeros
> >> 4) your integral is equal to the sum of f(...) / |grad(g(...))|
> evaluated
> >> at those zeros.
> >>
> >> I think that a nice explanation can be found in Appel. I can send a pdf
> >> with the book to you if you are interested.
> >>
> >>
> >> On 20 December 2011 19:45, Tom Bachmann <[email protected]> wrote:
> >>>
> >>> I think there are two basic cases to consider:
> >>>
> >>> 1) delta functions with *linear* arguments.
> >>>
> >>> This is (I believe) by a large margin the most common case. For this
> >>> deltaintegrate should simply be fixed; one only needs to correctly
> implement
> >>> integral(f(x) delta(a*x+b), (x, c, d)) and this shouldn't be hard.
> (Note
> >>> that this will also yield the multi-dimensional cases, by repeated
> >>> integration.)
> >>>
> >>> 2) delta functions with more complicated arguments
> >>>
> >>> Getting this to work is (I believe) much more work. It is non-trivial
> to
> >>> even define this.
> >>>
> >>> Tom
> >>>
> >>>
> >>> On 20.12.2011 16:21, Matthew Rocklin wrote:
> >>>>
> >>>> Hi Everyone,
> >>>>
> >>>> I'd like to compute multivariate integrals that contain Dirac
> >>>> Deltafunctions. I.e. expressions like
> >>>>
> >>>> integrate(exp(-(x**2+y**2))/pi * delta(2*x+3*y), (x,-oo, oo), (y,-oo,
> >>>> oo))
> >>>>
> >>>> The deltaintegrate function inside sympy fails to compute these
> >>>> correctly, see issue 2630.
> >>>> <http://code.google.com/p/sympy/issues/detail?id=2630>
> >>>>
> >>>> Wikipedia says
> >>>>
> >>>> <
> http://en.wikipedia.org/wiki/Dirac_delta_functions#Properties_in_n_dimensions
> >
> >>>>
> >>>> that you can compute general expressions of this form as follows:
> >>>>
> >>>> \int_{\mathbb{R}^n} f(\mathbf{x}) \, \delta(g(\mathbf{x})) \,
> >>>> d\mathbf{x} =
> >>>>
> >>>>
> \int_{g^{-1}(0)}\frac{f(\mathbf{x})}{|\mathbf{\nabla}g|}\,d\sigma(\mathbf{x})
> >>>>
> >>>> (hopefully the above image makes it through, if not go here)
> >>>>
> >>>>
> http://upload.wikimedia.org/wikipedia/en/math/3/3/f/33fbbed28ec715257d268faefc9e0e9f.png
> >>>>
> >>>> How hard would it be to compute the right hand side in sympy? In
> >>>> particular I'm confused by how to express the domain and what they
> mean
> >>>> by \delta\sigma(x)
> >>>>
> >>>> Or, if there is a better way of going about this I'm happy to hear it.
> >>>>
> >>>> -Matt
> >>>>
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