Green's Theorem and Conservative Fields

[n1person]

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.

 

[hamster143]

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.

 

[LCKurtz]

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...

 

[hamster143]

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.