Singularities and conservative vector fields

In summary, we discussed conservative vectorfields and their behavior around singularities. We learned that if a vectorfield has a singularity at the origin and its curl is zero, it is conservative in every open, simply connected subset in R^2 not containing the origin. However, the curve integral around a closed curve surrounding the origin may not always be zero. We also discussed the concept of constructing a line between two paths enclosing a singularity and taking the limit as the paths approach each other, resulting in a zero integral. This applies as long as the paths do not cross any singularities. This also applies to calculating the curve integral of a square around the singularity by adding the curve integrals along the four sides. Finally, we discussed
  • #1
NewGuy
9
0
I have a question regarding conservative vectorfields and singularities.
Suppose we have a vectorfield who is defined everywhere in R^2 except at the origin where it has a singularity, and suppose it's curl is zero. We then have that it is conservative in every open, simply connected subset in R^2 not containing the origin.
Now if we have a closed curve around the origin we can't generally say that the curve integral is zero right? That depends on the given vectorfield? If we have calculated the curve integral around a circle with centre in the origin and shown that it is zero, can we then say that every closed curve around the origin has curve integral zero?
Suppose for example we have a square around the origin. Can we calculate the curveintegral for the whole square by adding the curveintegrals along the 4 sides?
 
Mathematics news on Phys.org
  • #2
Yes, if you have two distinct paths, both inclosing a singularity, then you can construct a line from one (call it Ca) to the other (CB) and just a slight distance away, a second line back to the first singularity Consider the integration from any point on CA, counterclockwise around CA to the first "cutoff", along it to CB, around CB, clockwise to the second "cutoff", along it back to CA, and finally around CA to the starting point. We know have integrals: IA' is almost IA,the integral around A- it does not include that small section of A between the two "cutoffs". I1 is the integral along the first "cutoff" from CA to CB. IB' is almost -IB, the negative of the integral around CB- it does not include the small section between the "cutoffs" (negative because we have to go clockwise around B). Finally, I2 is the integral along the second "cutoff" from CB back to CA. Because there in no singularity inside that path the integral is 0: IA'+ I1- IB'+ I2= 0.

Now, take the limit as we move the two "cutoffs" toward each other. Since, in the limit, they become the same path but integrated in opposite directions, in the limit, I1[/sub[+ I2= 0. Also, the left out sections between the two "cutoffs" disappear: in the limit IA'= IA, the integral all the way around A, and IB'= IB, the integral all the way around B. Since there is no singularity between the two cutoffs, the left side of the equation remains 0. In the limit, IA- IB= 0 so IA= IA.

As long as, moving one closed path to another, we do not cross any singularities, the integrals are the same.

The last part, "Suppose for example we have a square around the origin. Can we calculate the curveintegral for the whole square by adding the curveintegrals along the 4 sides?" is a completely different question but, of course, it is true. That's just saying that the sum of the two partial integrals is equal to the entire integral. I would think you did that back in basic calculus. Did you never do a path integral of a broken line?
 
Last edited by a moderator:
  • #3
As long as, moving one closed path to another, we do not cross any singularities, the integrals are the same.

This is a nice thing indeed! If we have that the curve integral around the singularity is 2 pi that is true for all closed paths around the singularity? I suspect that it is a requirement that curl(F) is zero everywhere along the path? Does the same thing apply to R^3?

Regarding the square: I meant a square around the singularity. I know that in a curve integral you can add the different paths together, I just didn't know if you could around a singularity.

So if we somehow can show that a vector field is the gradient of some function (by guessing at an appropriate function for example), it is true that the curve integral is zero for all closed paths, regardless if they surround the singularity?
 

What is a singularity in a vector field?

A singularity in a vector field is a point where the magnitude or direction of the vector becomes undefined or infinite. This can occur when the vector field has a point of discontinuity or when the vector field is not differentiable at that point. Singularities are important to study because they can provide insight into the behavior of the vector field.

What are conservative vector fields?

Conservative vector fields are vector fields in which the line integral between any two points is independent of the path taken between those points. In other words, the work done by the vector field is only dependent on the initial and final positions, not the specific path taken. This is equivalent to saying that the curl of the vector field is zero, or that the vector field is irrotational.

What is the relationship between singularities and conservative vector fields?

In conservative vector fields, singularities can only occur at points where the magnitude of the vector field is zero. This is because if the vector field is conservative, the magnitude of the vector must be continuous throughout the field. Therefore, singularities in conservative vector fields are points where the direction of the vector becomes undefined, but the magnitude remains zero.

How are singularities and conservative vector fields used in physics?

In physics, singularities in vector fields can occur in many different phenomena, such as fluid dynamics, electromagnetism, and gravitational fields. The study of singularities and conservative vector fields allows scientists to better understand the behavior of these physical systems and make predictions about their behavior. In particular, conservative vector fields play a crucial role in understanding and modeling the behavior of conservative forces, such as gravity and electrostatic forces.

What are some real-world examples of singularities and conservative vector fields?

One example of a singularity in a conservative vector field is a point mass in a gravitational field. At the center of the point mass, the magnitude of the gravitational field is zero, and the direction becomes undefined. Another example is the electric field around a point charge, where the electric field strength becomes infinite at the location of the charge. Both of these examples illustrate the relationship between singularities and conservative vector fields, as well as their importance in understanding physical phenomena.

Similar threads

I Complex analysis - removable singular points
    • General Math
Replies
5
Views
2K
I Integral curves of (timelike) smooth vector field
    • Special and General Relativity
Replies
26
Views
1K
A Finding Minimal Mean Distance Curves on the Unit Sphere
    • General Math
Replies
2
Views
1K
A The abc conjecture and Virasoro algebra
    • General Math
Replies
1
Views
1K
I Some questions in "Introduction to quantum mechanics"
    • Quantum Physics
Replies
1
Views
664
B Gradient of scalar field is zero everywhere given boundary conditions
    • General Math
Replies
3
Views
936
A Do Black Holes have Singularities?
    • Special and General Relativity
Replies
20
Views
806
I Electric field vector takes into account the field's radial direction?
    • Electromagnetism
Replies
3
Views
698
A Parallel transport of a 1-form aound a closed loop
    • Special and General Relativity
Replies
8
Views
1K
A Connection between QFT renormalization and field regularization?
    • Quantum Physics
Replies
28
Views
1K

Hot Threads

Recent Insights

Back
Top