Use Stokes’ Theorem to evaluate ∫cF · dr, where

$$F(x,y,z) = x^2yi + 1/3x^3j+xyk$$

and C is the curve of intersection of the hyperbolic paraboloid

$$z=y^2-x^2$$

and the cylinder

$$x^2+y^2=1$$

oriented counterclockwise as viewed from above.

Use Stokes’ Theorem to evaluate integral through C F.dr. In each case C is oriented counterclockwise as viewed from above. F(x+y+z)=i+(x+yz)j+(xy-z^1/2)k, C is the boundary of the part of the plane 3x+2y+z=1 in the first octant

Use Stokes’ Theorem to evaluate ∫∫5 curl F · dS.

$$F(x, y, z) = x^2z^2i + y^2z^2j + xyzk$$

S is the part of the paraboloid

$$z=x^2+y^2$$

that lies inside the cylinder

$$x^2+y^2=4$$

, oriented upward

Use Stokes' Theorem to evaluate $\int_C \mathrm{F} \cdot d \mathrm{r}$, where $\mathrm{F}(x, y, z)=x y \mathrm{i}+y z \mathrm{j}+z x \mathrm{k}$, and $C$ is the triangle with vertices $(1,0,0),(0,1,0)$, and $(0,0,1)$, oriented counterclockwise as viewed from above.

Use Stokes' Theorem to evaluate $\int_C \mathrm{F} \cdot d \mathrm{r}$. In each case $C$ is oriented counterclockwise as viewed from above. Let

$$\mathrm{F}(x, y, z)=y z \mathrm{i}+2 x z \mathrm{j}+e^{x y} \mathrm{k}$$

where $C$ is the circle $x^2+y^2=16, z=5$.

Use Stokes’ Theorem to evaluate ∫∫s curl F · dS. F(x, y, z) = 2y cos zi + e^x sin zj + xe^y k, S is the hemisphere

$$x^2+y^2+z^2=9, z≥0,$$

oriented upward

Use Stokes’ theorem to evaluate $\iint_{S}(\operatorname{curl} \mathbf{F} \cdot \mathbf{N}) d S$ for the vector fields and surface. Use a CAS and Stokes’ theorem to evaluate $\iint_{S}(\operatorname{curl} \mathbf{F} \cdot \mathbf{N}) d S$, 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.

Use Stokes’ Theorem to evaluate surface integral curl F.ds F(x,y,z)=xyzi+xyj+x^2yzk, S consists of the top and the four sides (but not the bottom) of the cube with vertices (+-1, +-1, +-1), oriented outward

Use Stokes’ Theorem to evaluate integral C F.dr. In each case C is oriented counterclockwise as viewed from above. F(x.y,z)=(x+y^2)i+(y+z^2)j+(z+x^2)k, C is the triangle with vertices (1, 0, 0), (0, 1, 0), and (0, 0, 1)

Use Stokes’s Theorem to evaluate ∫_c F·dr. In each case, C is oriented counterclockwise as viewed from above. F(x, y, z) = 2yi + 3zj + xk C: triangle with vertices (2, 0, 0), (0, 2, 0), and (0, 0, 2)

888Use Stokes’ Theorem to evaluate double integral S curl F.dS. F(x,y,z)=e^xyi+e^xzj+x^zk, S is the half of the ellipsoid 4x^2+y^2+z^2=4 that lies to the right of the xz-plane, oriented in the direction of the positive y-axis

F(x, y, z) = xy i + 2zj + 3yk, C is the curve of intersection of the plane x + z = 5 and the cylinder

$$x^2+y^2 = 9$$

Use Stokes' Theorem to evaluate $\oint_Ce^x\cos z\ dx + 2xy^2\ dy + \cot^{-1}y\ dz$, where C is the circle x squared + y squared = 4 and z = 0. Hint: Find a surface S with C as its boundary and such that C is oriented counterclockwise when viewed from above.

Consider the surface integral $\iint_S(\operatorname{curl} \mathbf{F}) \cdot \mathbf{n} d S$, where $\mathbf{F}=x y z \mathbf{k}$ and $S$ is that portion of the paraboloid $z=1-x^2-y^2$ for $z \geq 0$ oriented upward. (a) Evaluate the surface integral by the method of section; that is, do not use Stokes' theorem. (b) Evaluate the surface integral by finding a simpler surface that is oriented upward and has the same boundary as the paraboloid. (c) Use Stokes' theorem to verify the result in part (b).