6.7 Stokes’ theorem  (Page 8/10)

[T] Use a CAS and Stokes’ theorem to evaluate ${\displaystyle {\iint }_S(\text{curl}\text{F}·\text{N})dS},$ where $\text{F}(x,y,z)=x^{2}y\text{i}+x{y}^2\text{j}+z^3\text{k}$ and C is the curve of the intersection of plane $3x+2y+z=6$ and cylinder $x^2+y^2=4,$ oriented clockwise when viewed from above.

[T] Use a CAS and Stokes’ theorem to evaluate ${\displaystyle \underset{S}{\iint }\text{curl}\text{F}·d\text{S},}$ where $\text{F}(x,y,z)=\left(\text{sin}\left(y+z\right)-y{x}^2-\frac{y^3}{3}\right)\text{i}+x\text{cos}\left(y+z\right)\text{j}+\text{cos}\left(2y\right)\text{k}$ and S consists of the top and the four sides but not the bottom of the cube with vertices $\left(\mathrm{\pm 1},\mathrm{\pm 1},\mathrm{\pm 1}\right),$ oriented outward.

[T] Use a CAS and Stokes’ theorem to evaluate ${\displaystyle \underset{S}{\iint }\text{curl}\text{F}·d\text{S},}$ where $\text{F}(x,y,z)=z^2\text{i}-3xy\text{j}+x^3{y}^3\text{k}$ and S is the top part of $z=5-x^2-y^2$ above plane $z=1,$ and S is oriented upward.

Use Stokes’ theorem to evaluate ${\displaystyle {\iint }_S(\text{curl}\text{F}·\text{N})dS},$ where $\text{F}(x,y,z)=z^2\text{i}+y^2\text{j}+x\text{k}$ and S is a triangle with vertices (1, 0, 0), (0, 1, 0) and (0, 0, 1) with counterclockwise orientation.

Use Stokes’ theorem to evaluate line integral ${\displaystyle \underset{C}{\int }\left(zdx+xdy+ydz\right)},$ where C is a triangle with vertices (3, 0, 0), (0, 0, 2), and (0, 6, 0) traversed in the given order.

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

Use Stokes’ theorem to evaluate ${\displaystyle {\iint }_S(\text{curl}\text{F}·\text{N})dS},$ where $\text{F}(x,y,z)=x\text{i}+y^2\text{j}+z{e}^{xy}\text{k}$ and S is the part of surface $z=1-x^2-2y^2$ with $z\ge 0\text{,}$ oriented counterclockwise.

Use Stokes’ theorem for vector field $\text{F}(x,y,z)=z\text{i}+3x\text{j}+2z\text{k}$ where S is surface $z=1-x^2-2y^2,z\ge 0,$ C is boundary circle $x^2+y^2=1,$ and S is oriented in the positive z -direction.

Use Stokes’ theorem for vector field $\text{F}(x,y,z)=-\frac{3}{2}{y}^2\text{i}-2xy\text{j}+yz\text{k}\text{,}$ where S is that part of the surface of plane $x+y+z=1$ contained within triangle C with vertices (1, 0, 0), (0, 1, 0), and (0, 0, 1), traversed counterclockwise as viewed from above.

A certain closed path C in plane $2x+2y+z=1$ is known to project onto unit circle $x^2+y^2=1$ in the xy -plane. Let c be a constant and let $\text{R}(x,y,z)=x\text{i}+y\text{j}+z\text{k}.$ Use Stokes’ theorem to evaluate ${\displaystyle {\int }_C^{}(c\text{k}\times \text{R})·d\text{S}}.$

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

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 ${\displaystyle {\iint }_S\text{curl}\text{F}·d\text{S}}.$

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

Use Stokes’ theorem to evaluate ${\displaystyle {\oint }_{}\text{F}·d\text{S}},$ 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 $(0,\mathrm{-2},2)$ oriented counterclockwise when viewed from above.

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 ${\displaystyle {\iint }_S\text{curl}\text{F}·d\text{S}},$ 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 counterclockwise.

Use Stokes’ theorem to evaluate ${\displaystyle {\iint }_S\text{curl}\text{F}·d\text{S}},$ where $\text{F}(x,y,z)=\text{−}{y}^2\text{i}+x\text{j}+z^2\text{k}$ and S is the part of plane $x+y+z=1$ in the positive octant and oriented counterclockwise $x\ge 0\text{,}y\ge 0\text{,}z\ge 0.$

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 counterclockwise and let F be a vector field with component functions that have continuous partial derivatives.

Let S be the part of paraboloid $z=9-x^2-y^2$ with $z\ge 0.$ 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 ${\displaystyle {\oint }_C\text{F}·d\text{S}}\text{,}$ if $\text{F}(x,y,z)=\left(3z-\text{sin}x\right)\text{i}+\left(x^2+e^y\right)\text{j}+\left(y^3-\text{cos}z\right)\text{k}\text{,}$ where C is the curve given by $x=\text{cos}t,y=\text{sin}t,z=1;0\le t\le 2\pi .$

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

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

Let S be paraboloid $z=a\left(1-x^2-y^2\right),$ for $z\ge 0,$ where $a>0$ is a real number. Let $\text{F}=⟨x-y,y+z,z-x⟩.$ For what value(s) of a (if any) does ${\displaystyle {\iint }_S\left(\nabla \times \text{F}\right)}·\text{n}dS$ have its maximum value?

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

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 $\text{F}=⟨2z+y,2x+z,2y+x⟩.$ Evaluate ${\displaystyle {\iint }_S\left(\nabla \times \text{F}\right)}·\text{n}dS.$

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

What is the circulation of C of vector field $\text{F}=⟨\text{−}y,\text{−}z,x⟩$ as a function of $\phi ?$

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

Circle C in plane $x+y+z=8$ has radius 4 and center (2, 3, 3). Evaluate ${\displaystyle {\oint }_C\text{F}·d\text{r}}$ for $F=⟨0,\text{−}z,2y⟩,$ where C has a counterclockwise orientation when viewed from above.

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

Evaluate integral ${\displaystyle {\iint }_S\left(\nabla \times \text{F}\right)}·\text{n}dS,$ where $\text{F}=\text{−}xz\text{i}+yz\text{j}+xy{e}^z\text{k}$ 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 .

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

$\text{F}=⟨y^2{z}^3,z2xy{z}^3,3x{y}^2{z}^2⟩$