gomunkul51 said:Correction: The line integral around any closed curve will be 0 ONLY if the integral is on a conservative field. otherwise it won't be zero for all cases.
Dick said:Mmm. Well, sure. F=(sin(y),x*cos(y)) is conservative. So the integral of F.dr is zero around any closed curve. I didn't mean ANY F. Did that really need a 'correction'?
Green's Theorem is a fundamental theorem in mathematics that relates the line integral of a two-dimensional vector field over a closed curve to the double integral of its divergence over the region enclosed by the curve.
Green's Theorem is used to calculate the values of line integrals and double integrals in a more efficient way, as it allows us to switch between the two types of integrals. It is also used in many applications such as fluid dynamics, electromagnetism, and heat transfer.
The significance of zero in Green's Theorem is that it states that the line integral of a conservative vector field over a closed curve is equal to zero. This means that the work done by the vector field in a closed path is zero, indicating that the field is conservative.
Green's Theorem is closely related to the concept of a conservative field since it states that a vector field is conservative if and only if the line integral of that field over a closed path is equal to zero. This is because conservative fields have a property known as path independence, meaning that the work done by the field is independent of the path taken.
Yes, Green's Theorem can be extended to higher dimensions through the use of the generalization known as the Stokes' Theorem. This theorem relates the integral of a differential form over an oriented manifold to the integral of its exterior derivative over the boundary of the manifold, and it encompasses Green's Theorem as a special case.