스토크스 정리

Use Stokes’ theorem to evaluate $\int_{C}[2x{y}^{2}zdx+2{x}^{2}yzdy+({x}^{2}{y}^{2}-2z)dz],$ where C is the curve given by $x=\cos \,t,y=\sin \,t,z=\sin \,t,0\le t\le 2\pi ,$ traversed in the direction of increasing t.

Use Stokes’ theorem to evaluate $\int_{C}\left(\frac{1}{2}{y}^{2}dx+zdy+xdz\right),$ where C is the curve of intersection of plane $x+z=1$ and ellipsoid ${x}^{2}+2{y}^{2}+{z}^{2}=1,$ oriented clockwise from the origin.

Let $\text{F}(x,y,z)=xy\text{i}+({e}^{{z}^{2}}+y)\text{j}+(x+y)\text{k}$ and let S be the graph of function $y=\frac{{x}^{2}}{9}+\frac{{z}^{2}}{9}-1$ with $y\le 0$ oriented so that the normal vector of S has a positive j component. Use Stokes’ theorem to compute integral ${\iint }_{S}\text{curl}\,\text{F}\cdot d\text{S}.$

Use the surface integral in Stokes’ theorem to calculate the circulation of field F, $\text{F}(x,y,z)={x}^{2}{y}^{3}\text{i}+\text{j}+z\text{k}$ around C, which is the intersection of cylinder ${x}^{2}+{y}^{2}=4$ and hemisphere ${x}^{2}+{y}^{2}+{z}^{2}=16,z\ge 0,$ oriented counterclockwise when viewed from above.

Use Stokes’ theorem to compute ${\iint }_{S}\text{curl}\,\text{F}\cdot d\text{S},$ where $\text{F}(x,y,z)=\text{i}+x{y}^{2}\text{j}+x{y}^{2}\text{k}$ and S is a part of plane $y+z=2$ inside cylinder ${x}^{2}+{y}^{2}=1$ and oriented upward.

Take paraboloid $z={x}^{2}+{y}^{2},$ for $0\le z\le 4,$ and slice it with plane $y=0.$ Let S be the surface that remains for $y\ge 0,$ including the planar surface in the xz-plane. Let C be the semicircle and line segment that bounded the cap of S in plane $z=4$ with counterclockwise orientation. Let $\text{F}=\langle 2z+y,2x+z,2y+x\rangle .$ Evaluate ${\iint }_{S}\text{curl} \text{F}\cdot dS$