Vector analysis and distributions

In summary, the conversation is discussing the values of delta and divergence in various cases. It is mentioned that in general, the value of delta for 1/r is -4pi times the delta distribution at the point r. The conversation then delves into the calculation of div(r/r^3) and how it diverges at the origin. The use of the divergence theorem is suggested to compute the delta distribution at the origin.
  • #1
LagrangeEuler
717
20
In many books it is just written that ##\Delta(\frac{1}{r})=0##. However it is only the case when ##r \neq 0##. In general case ##\Delta(\frac{1}{r})=-4\pi \delta(\vec{r})##. What abot ##\mbox{div}(\frac{\vec{r}}{r^3})##? What is that in case where we include also point ##0##?
 
  • Like
Likes Delta2
Physics news on Phys.org
  • #2
Direct calculation gives ##-2r^{-3}## if I do it right. it diverges at the Origin.
 
  • #3
You can use the divergence theorem to compute what delta distribution it is around the origin. When you compute the surface integral you get a constant value no matter which surface you use (you can verify this with spheres pretty easily) and that implies a delta distribution at the origin. You can make this more rigorous depending on how you define a delta distribution
 

Related to Vector analysis and distributions

1. What is vector analysis?

Vector analysis is a branch of mathematics that deals with the study of vectors, which are mathematical quantities that have both magnitude and direction. It involves operations such as addition, subtraction, and scalar multiplication of vectors, as well as concepts like dot and cross products.

2. How is vector analysis used in science?

Vector analysis is used in various fields of science, including physics, engineering, and computer science. It is used to represent physical quantities such as force, velocity, and acceleration, and to solve problems involving these quantities. It is also used in computer graphics to represent and manipulate 3D objects.

3. What are vector distributions?

Vector distributions, also known as vector fields, are mathematical functions that assign a vector to each point in a given space. They are used to represent physical quantities that vary in space, such as wind velocity, magnetic fields, and electric fields. Vector analysis is used to study and analyze these distributions.

4. What are some common applications of vector analysis and distributions?

Some common applications of vector analysis and distributions include calculating the motion of objects in physics, designing and analyzing electrical circuits in engineering, and creating realistic computer-generated images in computer graphics. It is also used in geology to study the flow of fluids in porous materials and in meteorology to analyze weather patterns.

5. What are some important theorems in vector analysis?

Some important theorems in vector analysis include the divergence theorem, which relates the flux of a vector field through a closed surface to the divergence of the field within the surface, and the curl theorem, which relates the circulation of a vector field around a closed curve to the curl of the field within the curve. Other important theorems include the gradient theorem and the Green's theorem.

Similar threads

  • Calculus
Replies
4
Views
2K
Replies
4
Views
652
Replies
11
Views
2K
Replies
2
Views
1K
Replies
5
Views
1K
Replies
1
Views
1K
Replies
2
Views
509
Replies
16
Views
3K
  • Calculus
Replies
9
Views
2K
Back
Top