Use Stokes' Theorem to evaluate $\int_{C} \mathbf{F} \cdot d \mathbf{r},$ where

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

and C is the curve of intersection of the plane x + y + z = 1 and the cylinder $x^{2}+y^{2}=9$ oriented counterclockwise as viewed from above.

Use Stokes’ theorem to evaluate $\oint_{C} \mathbf{F} \cdot d \mathbf{r}$. Assume C is oriented counterclockwise as viewed from above. $\mathbf{F}=(2 z+x) \mathbf{i}+(y-z) \mathbf{j}+(x+y) \mathbf{k}$; C the triangle with vertices $(1,0,0),(0,1,0),(0,0,1)$.

Integrate the given function over the given surface. H(x, y, z) = yz, over the part of the sphere

$$x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4$$

that lies above the cone

$$z = \sqrt { x ^ { 2 } + y ^ { 2 } }.$$

Use Stokes’ theorem to evaluate $\iint_{S}(\operatorname{curl} \mathbf{F} \cdot \mathbf{N}) d S$ for the vector fields and surface. Use Stokes’ theorem to evaluate line integral $\int_{C}(z d x+x d y+y d z)$, 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 ∫c F · dr, where F(x, y, z) = xyi + yzj + zxk, and C is the triangle with vertices (1, 0, 0), (0, 1, 0), and (0, 0, 1), oriented counter clockwise as viewed from above.

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.

Sketch the curve C with parametric equations

$$x=cos t, y=sin t, z=sin t, 0≤t≤2π$$

Use Stokes’ Theorem to evaluate ∫cF · dr. In each case C is oriented counterclockwise as viewed from above. F(x, y, z) = yzi+2xzj+e^xy k, C is the circle

$$x^2+y^2=16, z = 5$$

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 $\iint_{S}(\operatorname{curl} \mathbf{F} \cdot \mathbf{N}) d S$ for the vector fields and surface. $\iint_{S} \operatorname{curl} \mathbf{F} \cdot d \mathbf{S}$, and S consists of the top and the four sides but not the bottom of the cube with vertices $(\pm 1, \pm 1, \pm 1)$, oriented outward.

Use Stokes' Theorem to evaluate $\iint_{S} \operatorname{curl} \mathbf{F} \cdot d \mathbf{S}.$ $\mathbf{F}(x, y, z)=x y z\ \mathbf{i}+x y\ \mathbf{j}+x^{2} y z\ \mathbf{k},$ S consists of the top and the four sides (but not the bottom) of the cube with vertices $(\pm 1,\pm 1,\pm 1),$ oriented outward

Find the volume of the solid in the first octant bounded by the cylinder $z=9-y^2$ and the plane $x=2$.

Find a parametric representation for the surface.

The part of the sphere $x^2+y^2+z^2=4$ that lies above the cone $z=\sqrt{x^2+y^2}$

Find a parametric representation for the surface. The part of the sphere $x^2+y^2+z^2=4$ that lies above the plane $z=\sqrt{2}$.