Green's Theorem in 3 Dimensions for non-conservative field

In summary, The given conversation discusses finding the integral of a non-conservative vector field over a directed curve forming a triangle in three-dimensional space. The curl theorem and Green's identities are mentioned as possible approaches, but it is ultimately concluded that the integral can be calculated directly using the parametrization of the curve.
  • #1
Tom31415926535
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Homework Statement


C is the directed curve forming the triangle (0, 0, 0) to (0, 1, 1) to (1, 1, 1) to (0, 0, 0).

Let F=(x,xy,xz) Find ∫F·ds.

Homework Equations

The Attempt at a Solution


My intial instinct was to check if it was conservative. Upon calculating:

∇xF=(0,-z,y)

I concluded that it isn't conservative. How do I apply Green's theorem for a three-dimensional vector field that is non-conservative?
 
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  • #2
What you are after seems to be the curl theorem (also called Stokes’ theorem). It relates the circulation integral of a field with the integral over a surface of the curl of the field. Green’s identities relate surface integrals to volume integrals based on the divergence theorem.

That being said, I see no need to apply any integral theorem here. The three line integrals involved should be rather straightforward.
 
  • #3
The integral can easily be calculated directly
 

Related to Green's Theorem in 3 Dimensions for non-conservative field

What is Green's Theorem in 3 Dimensions for non-conservative field?

Green's Theorem in 3 Dimensions for non-conservative field is a mathematical theorem that relates the line integral of a vector field over a closed curve to the double integral of the curl of the vector field over the surface bounded by the curve. It is an extension of the 2-dimensional Green's Theorem to 3 dimensions.

What is the significance of Green's Theorem in 3 Dimensions for non-conservative field?

Green's Theorem in 3 Dimensions for non-conservative field is significant because it allows for the calculation of surface integrals in three dimensions using simpler line integrals. It also helps in determining if a given vector field is conservative or not, as conservative fields follow a different version of Green's Theorem.

How is Green's Theorem in 3 Dimensions applied in real-world scenarios?

Green's Theorem in 3 Dimensions for non-conservative field has various applications in physics, engineering, and other fields. It is used to calculate work done by non-conservative forces, such as friction, in three-dimensional systems. It is also used in fluid dynamics to calculate fluid flow and circulation around a closed curve.

What are the limitations of Green's Theorem in 3 Dimensions for non-conservative field?

Green's Theorem in 3 Dimensions for non-conservative field has limitations when applied to highly irregular or non-smooth surfaces, as it relies on the assumption of a continuous and differentiable vector field. It also cannot be applied to surfaces with holes or boundaries that intersect themselves.

Are there any other variations of Green's Theorem besides the 3-dimensional version for non-conservative field?

Yes, besides the 3-dimensional version for non-conservative field, there are also 2-dimensional versions for both conservative and non-conservative fields. There are also higher-dimensional versions for more complex systems. Additionally, there are alternative theorems, such as Stokes' Theorem and the Divergence Theorem, that are closely related to Green's Theorem and have their own variations.

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