Let F(x,y)=⟨1,−xy⟩ and C be the unit circle centered at the origin drawn in a counterclockwise fashion as t runs from 0 to 2π. Set up an integral that will compute the flow along a curve C (circulation) and an integral that will compute the flow across a curve C (flux).
I'm confused on what is meant by "circulation"... is this interchangeable with scalar curl? Also, it should be given in this format for "circulation": ∮F∙dp (across C) = ∮(____)dt from _ to _ and in this format for flux: ∮F∙dn (across C) = ∮(____)dt from _ to _
What is p? What is n?
I figured that the limits for both are 0 to 2pi, but I don't understand what to put inside the actual integral for either, especially circulation. For flux, I tried just putting in the divergence of F (by the Divergence Theorem) but that didn't work because it's in terms of dt and not dV.
Any help is appreciated!
If $C$ is a simple, oriented, closed curve in the plane with unit tangent field $\mathbf{T}$ and outward unit normal vector field $\mathbf{n}$, and $\mathbf{F}$ is a vector field on $C$, the circulation of $\mathbf{F}$ around $C$ is the integral $$ \int_C \mathbf{F}\cdot \mathbf{T}\,ds = \int_C \mathbf{F}\cdot d\mathbf{s} $$ The circulation is related to the scalar curl by Green's theorem.
The flux of $\mathbf{F}$ across $C$ is the integral $$ \int_C \mathbf{F} \cdot \mathbf{n}\,ds $$ The flux is related to the divergence by Gauss's theorem.