벡터 미적분 전체 문제

Use Stokes’ theorem and let C be the boundary of surface $z={x}^{2}+{y}^{2}$ with $0\le x\le 2$ and $0\le y\le 1,$ oriented with upward facing normal. Define

$$\text{F}(x,y,z)=[\sin ({x}^{3})+xz]\text{i}+(x-yz)\text{j}+\cos ({z}^{4})\text{k}\,\text{and evaluate}\,{\int }_{C}\text{F}\cdot d\text{r}.$$

Let S be hemisphere ${x}^{2}+{y}^{2}+{z}^{2}=4$ with $z\ge 0,$ oriented upward. Let $\text{F}(x,y,z)={x}^{2}{e}^{yz}\text{i}+{y}^{2}{e}^{xz}\text{j}+{z}^{2}{e}^{xy}\text{k}$ be a vector field. Use Stokes’ theorem to evaluate ${\iint }_{S}\text{curl}\,\text{F}\cdot d\text{S}.$

Use Stokes’ theorem to evaluate ${\int }_{C}F\cdot dr$ where $\text{F}(x,y,z)=y\text{i}+z\text{j}+x\text{k}$ and C is a triangle with vertices (0, 0, 0), (2, 0, 0) and $(0,-2,2)$ oriented counterclockwise when viewed from above.

Use Stokes’ theorem to evaluate ${\iint }_{S}\text{curl}\,\text{F}\cdot d\text{S},$ where $\text{F}(x,y,z)=-{y}^{2}\text{i}+x\text{j}+{z}^{2}\text{k}$ and S is the part of plane $x+y+z=1$ in the first octant and oriented upward $x\ge 0,\,y\ge 0,\,z\ge 0.$

Let $\text{F}(x,y,z)=xy\text{i}+2z\text{j}-2y\text{k}$ and let C be the intersection of plane $x+z=5$ and cylinder ${x}^{2}+{y}^{2}=9,$ which is oriented counterclockwise when viewed from the top. Compute the line integral of F over C using Stokes’ theorem.

[T] Use a CAS and let $\text{F}(x,y,z)=x{y}^{2}\text{i}+(yz-x)\text{j}+{e}^{yxz}\text{k}.$ Use Stokes’ theorem to compute the surface integral of curl F over surface S with inward orientation consisting of cube $[0,1]\,\times \,[0,1]\,\times \,[0,1]$ with the right side missing.

Let S be ellipsoid $\frac{{x}^{2}}{4}+\frac{{y}^{2}}{9}+{z}^{2}=1$ oriented outward and let F be a vector field with component functions that have continuous partial derivatives. Compute ${\iint }_{s}\text{curl }F\cdot dS$

Let S be the part of paraboloid $z=9-{x}^{2}-{y}^{2}$ with $z\ge 0$ oriented upward. Verify Stokes’ theorem for vector field $\text{F}(x,y,z)=3z\text{i}+4x\text{j}+2y\text{k}.$

[T] Use a CAS and Stokes’ theorem to evaluate ${\int }_{C}F\cdot dr$ if $\text{F}(x,y,z)=(3z-\sin \,x)\text{i}+({x}^{2}+{e}^{y})\text{j}+({y}^{3}-\cos \,z)\text{k},$ where C is the curve given by $x=\cos \,t,y=\sin \,t,z=1;0\le t\le 2\pi .$

[T] Use a CAS and Stokes’ theorem to evaluate ${\iint }_{S}\text{curl}\text{F}\cdot d\text{S},$ where $\text{F}(x,y,z)=2y\text{i}+{e}^{z}\text{j}-\arctan \,x\text{k}$ with S as a portion of paraboloid $z=4-{x}^{2}-{y}^{2}$ cut off by the xy-plane oriented upward.

[T] Use a CAS to evaluate ${\iint }_{S}\text{curl}\text{F}\cdot d\text{S},$ where $\text{F}(x,y,z)=2z\text{i}+3x\text{j}+5y\text{k}$ and S is the surface parametrically by $\text{r}(r,\theta )=r\,\cos \,\theta \text{i}+r\,\sin \,\theta \text{j}+(4-{r}^{2})\text{k}$ $(0\le \theta \le 2\pi ,\,0\le r\le 3).$

Let S be paraboloid $z=a(1-{x}^{2}-{y}^{2}),$ for $z\ge 0,$ where $a>0$ is a real number. Let $\text{F}=\langle x-y,y+z,z-x\rangle .$ For what value(s) of a (if any) does ${\iint }_{S}\text{curl} \text{F}\cdot dS$ have its maximum value?

Evaluate a surface integral over a more convenient surface to find the value of A.

For the following application exercises, the goal is to evaluate $A={\iint }_{S}\text{curl} \text{F}\cdot dS$ where $\text{F}=\langle xz,-xz,xy\rangle$ and S is the upper half of ellipsoid ${x}^{2}+{y}^{2}+8{z}^{2}=1,\,\text{where}\,z\ge 0.$

Evaluate A using a line integral.

What is the length of C in terms of $\varphi ?$

What is the circulation of C of vector field $\text{F}=\langle -y,-z,x\rangle$ as a function of $\varphi ?$

For what value of $\varphi$ is the circulation a maximum?

Circle C in plane $x+y+z=8$ has radius 4 and center (2, 3, 3). Evaluate ${\int }_{C}\text{F}\cdot d\text{r}$ for $F=\langle 0,-z,2y\rangle ,$ where C has a counterclockwise orientation when viewed from above.

Evaluate integral ${\iint }_{S}\text{curl} \text{F}\cdot dS$ where $\text{F}=-xz\text{i}+yz\text{j}+xy{e}^{z}\text{k}$ and S is the cap of paraboloid $z=5-{x}^{2}-{y}^{2}$ above plane $z=3,$ and n points in the positive z-direction on S.

For the following exercises, use Stokes’ theorem to find the circulation of the following vector fields around any smooth, simple closed curve C.

$\text{F}=\nabla (x\,\sin \,y{e}^{z})$