벡터 미적분 전체 문제

[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.

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 ${\iint }_{S}\text{curl} \text{F}\cdot dS$

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.

Velocity field $\text{v}=\langle 0,1-{x}^{2},0\rangle ,$ for $|x|\le 1\,\text{and}\,|z|\le 1,$ represents a horizontal flow in the y-direction. Compute the curl of v in a clockwise rotation.

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.

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

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

$\text{F}=\langle {y}^{2}{z}^{3},2xy{z}^{3},3x{y}^{2}{z}^{2}\rangle$

For the following exercises, use a computer algebraic system (CAS) and the divergence theorem to evaluate surface integral ${\iint }_{S}^{}\text{F}\cdot \text{N}ds$ for the given choice of F and the boundary surface S. For each closed surface, assume N is the outward unit normal vector.

[T] $\text{F}(x,y,z)=x\text{i}+y\text{j}+z\text{k};$ S is the surface of cube $0\le x\le 1,0\le y\le 1,0<z\le 1.$

[T] $\text{F}(x,y,z)=(\cos \,yz)\text{i}+{e}^{xz}\text{j}+3{z}^{2}\text{k};$ S is the surface of hemisphere $z=\sqrt{4-{x}^{2}-{y}^{2}}$ together with disk ${x}^{2}+{y}^{2}\le 4$ in the xy-plane.

[T] $\text{F}(x,y,z)=({x}^{2}+{y}^{2}-{x}^{2})\text{i}+{x}^{2}y\text{j}+3z\text{k};S$ is the surface of the unit cube $0\le x\le 1,0\le y\le 1,0\le z\le 1$ excluding the face $z=0$.

[T] $\text{F}(x,y,z)=x\text{i}+y\text{j}+z\text{k};$ S is the surface of the solid bounded by the parabola $z={x}^{2}+{y}^{2}\,$ and the plane $z=9$.

For the following exercises, use a computer algebraic system (CAS) and the divergence theorem to evaluate surface integral ${\iint }_{S}^{}\text{F}\cdot \text{N}ds$ for the given choice of F and the boundary surface S. For each closed surface, assume N is the outward unit normal vector.

[T] $\text{F}(x,y,z)={x}^{2}\text{i}+{y}^{2}\text{j}+{z}^{2}\text{k};$ S is the surface of sphere ${x}^{2}+{y}^{2}+{z}^{2}=4.$