Delta function in spherical coords

In summary, The delta function is a function that is zero if the 'r' variable is non-zero. It can be written in cylindrical or spherical coordinates.
  • #1
Lorna
45
0

Homework Statement


If we have a delta function in cartesian coords, how do we convert it into spherical.
for example : delta (r) = delta(x-x0) delta(y-y0) delta(z-z0)

Homework Equations





The Attempt at a Solution


I used
delta (r) = delta(r-r0) delta(cos{theta}-cos{theta0}) delta (phi-phi0)/(r sin{theta})^2


How do we find r0,cos{theta0} and phi0, if what I am using is the right formula.

do we use:

z= r cos{theta}
y= r sin{theta} cos {phi}
x= r sin {theta} sin {phi}

and then say the delta function is non-zero if x=x0 or x0=r sin {theta} sin {phi}
and so on and then solve fir r, cos{theta} and phi?

thanks
 
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  • #2
I'm not sure what you mean by writing it in spherical coordinates. The Delta function is 0 if [itex]\rho[/itex] (your r) is non-zero.
 
  • #3
If you were working in polar coordinates for example, the 'obvious' thing to do would be to write [tex]\delta = \delta \left( r \right)[/tex] but this is incorrect since the delta function would not satisfy all of the required properties. I can't remember exactly off the top of my head but in cylindrical coordinates you define the delta function as something like [tex]\delta \left( {x,y} \right) = \frac{{\delta \left( r \right)}}{{2\pi r}}[/tex].

I would imagine that the delta function would be defined in a similar way for spherical coordinates. My guess would be that the 2*pi*r would be replaced by 4*pi*r in spherical coordinates.
 
  • #5
Lorna, your solution is correct, but you should really write the left-hand side as
[tex]\delta(\vec{r}-\vec{r}_0)[/tex]
This is sometimes written as
[tex]\delta^3(\vec{r}-\vec{r}_0)[/tex]
to emphasize that this is a 3-dimensional delta function
 
Last edited:
  • #6
thanks all
 

1. What is the Delta function in spherical coordinates?

The Delta function in spherical coordinates, also known as the Dirac delta function, is a mathematical function that is defined as zero everywhere except at the origin, where it is infinitely tall and has an area of 1. It is commonly used in physics and engineering to represent point sources or to simplify certain mathematical expressions.

2. How is the Delta function expressed in spherical coordinates?

In spherical coordinates, the Delta function is expressed as δ(r), where r represents the radial distance from the origin. It can also be expressed using the coordinates (r, θ, φ) or (ρ, θ, φ), where θ and φ represent the polar and azimuthal angles, respectively.

3. What is the relationship between the Delta function and the volume element in spherical coordinates?

The Delta function in spherical coordinates is related to the volume element (r²sinθ) in the following way: δ(r) = 1/(r²sinθ). This relationship is important in evaluating integrals involving the Delta function in spherical coordinates.

4. How is the Delta function used in solving boundary value problems in spherical coordinates?

The Delta function is often used in solving boundary value problems in spherical coordinates. It can be used to represent point sources or to impose boundary conditions at a specific point in space. It can also be used to simplify certain differential equations by converting them into integral equations.

5. Are there any limitations to using the Delta function in spherical coordinates?

Yes, there are some limitations to using the Delta function in spherical coordinates. One limitation is that it can only be used to represent point sources, not extended sources. Additionally, it is a non-analytic function, which means it cannot be differentiated or integrated in the traditional sense. This can make it challenging to use in some mathematical expressions.

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