91Ó°ÊÓ

The curl of a two-dimensional vector field F(x, y) = (M(x, y), N(x, y)) is given by the scalar function: \(\text{curl} \ F(x, y) = \frac{\partial N}{\partial x} - \frac{\partial M}{\partial y}\) Circulation measures how much a vector field "circulates" around a closed curve C. It is the line integral of the vector field F around the curve C: \(\oint_C F \cdot d\textbf{r}\) Define the line integral of the vector field F along a curve: \(\int_C F \cdot d\textbf{r} = \int_C M dx + N dy\) Green's theorem relates the line integral of the vector field around a closed curve C to the double integral of the curl of the vector field over the region R enclosed by C. Specifically, \(\oint_C M dx + N dy = \iint_R (\frac{\partial N}{\partial x} - \frac{\partial M}{\partial y}) dA\)

We are given that the curl of the vector field F is zero in the region R: \(\frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} = 0\) Now, we apply Green's theorem to find the circulation on the closed curve C: \(\oint_C F \cdot d\textbf{r} = \oint_C M dx + N dy = \iint_R (\frac{\partial N}{\partial x} - \frac{\partial M}{\partial y}) dA\)

Since the curl of F is zero in the region R, we can write Green's theorem as: \(\oint_C F \cdot d\textbf{r} = \iint_R 0 \ dA\) The double integral of a constant (zero) function over a region R is equal to zero times the area of R, which is still zero: \(\iint_R 0 \ dA = 0\) Therefore, the circulation on the closed curve C is zero: \(\oint_C F \cdot d\textbf{r} = 0\) This concludes the proof that when a two-dimensional vector field has zero curl on a region, its circulation on a closed curve that bounds the region is zero.