Understanding Dirac Delta Function: Time Derivative & Hankel Transformation

In summary: No, I'm still not sure. Can you please be more explicity?In summary, the delta function is a distribution, and its time derivative is not differentiable in the classical sense. To consider differentiation, we have to revert to generalized derivatives and assume a certain level of differentiability on f.
  • #1
femiadeyemi
13
0
Hi All,
I have a problem in understanding the concept of dirac delta function. Let say I have a function, q(r,z,t) and its defined as q(r,z,t)= δ(t)Q(r,z), where δ(t) is dirac delta function and Q(r,z) is just the spatial distribution.
My question are:
1. How can I find the time derivative of this function, that is, [itex]\frac{\partial q(r,z,t)}{\partial t}[/itex]?
2. will hankel transformation of [itex]\frac{\partial q(r,z,t)}{\partial t}[/itex] be equal to zero (even when Q(r,z) [itex] \neq [/itex] 0)?

Thank you in advance.
FM
 
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  • #2
https://www.physicsforums.com/showthread.php?t=372548
The delta function is actually a distribution, and is not differentiable in the classical sense. In order to consider such differentiation, we have to revert to generalized derivatives. This is done by assuming a certain level of differentiability on f and some vanishing conditions.
-- Kreizhn (post #2)
 
  • #3
Hi Simon,
Thanks for your response. Unfortunately, I'm still not totally clear. Can you please be more explicity.
Once again, thank you.
FM
 
  • #4
Did you read the link?
 
  • #5
Yes, I did, but I didn't fully grasp it. Anyway, this is what I can come up with, please take a look and let me know if it makes (physical) sense.
Definition: q(r,z,t)=δ(t)Q(r,z)
[itex]\frac{\partial q(r,z,t)}{\partial t} = Q(r,z) \frac{d}{dt}[δ(t)][/itex]
[itex]\frac{\partial q(r,z,t)}{\partial t} = Q(r,z) δ^{'}(t)[/itex]
[itex]\frac{\partial q(r,z,t)}{\partial t} = Q(r,z) \frac{t}{t} δ^{'}(t)[/itex]
since: [itex]x δ^{'}(x) = -δ(x) [/itex]
Hence,
[itex]\frac{\partial q(r,z,t)}{\partial t} = -\frac{Q(r,z)}{t} δ(t)[/itex]
Thank you for your help
FM
 
  • #6
Did you understand it well enough to grasp what a "distribution" or "generalized function" is?
 

Related to Understanding Dirac Delta Function: Time Derivative & Hankel Transformation

1. What is the Dirac Delta Function?

The Dirac Delta Function, denoted as δ(x), is a mathematical function that is used to represent a point mass or impulse at a specific point in space. It is often described as a spike or spike-like function with an area of 1 under its curve.

2. What is the time derivative of the Dirac Delta Function?

The time derivative of the Dirac Delta Function, denoted as δ'(x), is a mathematical function that represents the rate of change of the Dirac Delta Function with respect to time. It is often used in physics and engineering to model impulses or sudden changes in a system over time.

3. How is the Dirac Delta Function related to the Heaviside Step Function?

The Dirac Delta Function and the Heaviside Step Function are closely related. The Heaviside Step Function, denoted as H(x), is the integral of the Dirac Delta Function. It is often used to represent a sudden switch or transition in a system at a specific point in time.

4. What is the Hankel Transformation?

The Hankel Transformation is a mathematical operation that converts a function of a single variable, such as the Dirac Delta Function, into a function of a different variable. It is commonly used in engineering and physics to analyze and solve problems involving wave propagation and diffraction.

5. How is the Dirac Delta Function used in real-world applications?

The Dirac Delta Function has many applications in physics, engineering, and mathematics. It is used to represent point sources or impulses in systems, such as in signal processing, circuit analysis, and quantum mechanics. It is also used in solving differential equations and in Fourier analysis to represent periodic functions as a sum of Dirac Delta Functions.

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