題目

9. (10 points) Let

F(x,y)=\dfrac{-2y}{x^2+y^2}\mathbf{i}+\dfrac{2x}{x^2+y^2}\mathbf{j}

and let

D=\{(x,y)\mid x^2+y^2=9\}

. Find

\int_{\partial D}\mathbf{F}\cdot \mathbf{T}\,ds,

where we traverse the boundary $\partial D$ in the counterclockwise direction and $\mathbf{T}$ is the unit tangent vector.

解答

令 $x=3\cos t$ 、 $y=3\sin t$ ， $0\le t\le2\pi$ 。逆時針單位切向量為

\mathbf T=(-\sin t,\cos t),

而 $\mathbf F=(-\frac23\sin t,\frac23\cos t)$ ，且 $\mathrm{d}s=3\,\mathrm{d}t$ 。故

\int_{\partial D}\mathbf F\cdot\mathbf T\,\mathrm{d}s =\int_0^{2\pi}2\,\mathrm{d}t =\boxed{4\pi}.