Green's Theorem and Conservative Fields

In summary: This means that the field behaves like a conservative field in that small area. Therefore, you can integrate along the line and find a scalar field that approximates the behavior of the vector field in that area. This scalar field will have a gradient that points in the direction of the vector field. So, in summary, the vector field x^2yi+xy^2j is not conservative, but locally along the y = x line it behaves like a conservative field. It is possible to find an approximate scalar field whose gradient behaves like the vector field in that small area.
  • #1
n1person
145
0
So let's say we have the vector field x^2yi+xy^2j, obviously the field is not conservative since dq/dx-dp/dy=y^2-x^2=/=0

however, let's say we wanted to find where locally the field would behave like a potential field, so we set y^2-x^2=0, so y=x (along the y=x line the field behaves like a conservative field). So my question is, a) is this true? b) is there some way to get an approximate scalar field whose gradiant behaves like the vector field locally along the y=x line?

Just something I was pondering.
 
Physics news on Phys.org
  • #2
Yes.

Since your field is parallel to y=x line at all points, you can integrate along the line and get values of the scalar field along that line, and then generalize it somehow so that its gradient points in the direction of the field.
 
  • #3
hamster143 said:
Yes.

Since your field is parallel to y=x line at all points, you can integrate along the line and get values of the scalar field along that line, and then generalize it somehow so that its gradient points in the direction of the field.

But his field isn't parallel to the y = x line. Not that I understand what you are getting at anyway...
 
  • #4
LCKurtz said:
But his field isn't parallel to the y = x line. Not that I understand what you are getting at anyway...

Locally along the y = x line it is.
 

Related to Green's Theorem and Conservative Fields

1. What is Green's Theorem?

Green's Theorem is a mathematical tool used to evaluate line integrals over a closed curve in the plane. It relates the line integral to a double integral over the region enclosed by the curve.

2. How is Green's Theorem used in physics and engineering?

Green's Theorem is used in physics and engineering to simplify calculations involving work done by a conservative force or the flux of a vector field through a closed surface. It allows for the conversion of a difficult line integral into an easier double integral.

3. What is a conservative field?

A conservative field is a vector field in which the line integral between any two points is independent of the path taken. In other words, the work done by a conservative force is only dependent on the initial and final positions, not the path taken.

4. How do you determine if a vector field is conservative?

A vector field is conservative if its curl is equal to zero. In other words, if the cross partial derivatives of the vector field are equal, then the field is conservative. Alternatively, if the line integral of the vector field over a closed curve is equal to zero, then the field is also conservative.

5. What are some real-world applications of Green's Theorem and conservative fields?

Green's Theorem and conservative fields have many applications in physics and engineering, such as in calculating the work done by a conservative force, determining the flow of fluid through a closed surface, and solving problems involving potential energy. They are also used in fields like electromagnetism, fluid dynamics, and thermodynamics.

Similar threads

Replies
4
Views
1K
Replies
4
Views
326
Replies
7
Views
2K
  • Calculus
Replies
3
Views
4K
  • Calculus and Beyond Homework Help
Replies
3
Views
579
  • Calculus and Beyond Homework Help
Replies
10
Views
684
Replies
3
Views
1K
  • Calculus
Replies
1
Views
1K
Back
Top