91影视

Use the Fundamental Theorem of Line Integrals, if applicable, to evaluate the integrals in Exercises 37鈥�44. Otherwise, show that the vector field is not conservative.

$\mathbf{F}(x,y,z)=(鈭�z,1,x)$, with C the circular helix given by $x=\cos t,y=t,z=\sin t,\mathrm{for}0鈮�t鈮�2\mathrm{蟺}$.

$\mathbf{F}(x,y,z)=8x\mathbf{i}-13y\mathbf{j}+\left(13z-12e^y\right)\mathbf{k}$, and $S$ is the surface of the region $W$ that lies above the region $\left|y\right|鈮�\left|x\right|,y鈮�0$ in the$xy$-plane, within the unit circle, and below the saddle $z=x^2-y^2$.

Directly compute (i.e., without using Green鈥檚 Theorem) ${鈭�}_R鈥�(3y鈭�3x)dA$, where R is the portion of the disk of radius 2, centered at the origin, and lying above the x-axis.

Show that the vector fields in Exercises 33鈥�40 are not conservative.

$\mathbf{G}(x,y,z)=(3,yz,z+12)$

$\mathbf{F}(x,y,z)=z\cos yz\mathbf{j}+z\sin yz\mathbf{k}$, where S is the portion of the plane with equation $2x鈭�8y鈭�10z=42$ that lies on the positive side of the rectangle with corners$\left(0,-\mathrm{蟺},0\right)(0,蟺,0),(0,蟺,蟺),(0,鈭�蟺,蟺)$in theyz-plane.

Arc length: Review the discussion of arc length in Sections 6.3 and 11.4. Compute the length of the following curves.

The plane curve $\left(t,t^2\right)$from $t=4\mathrm{to}t=6$

Green鈥檚 Theorem: Let $\mathit{F}(x,y)=\left(F_1\left(x,y\right),F_2\left(x,y\right)\right)$be a vector field defined on a region $R$in the $xy-$plane whose boundary is a ______ curve $C$. If $\mathit{r}\left(t\right)$is a parametrization of $C$in the counterclockwise direction (as viewed

from the positive $z-$axis), then

$$\begin{gathered} {鈭�}_C\mathit{F}(x,y)路dx={鈭�}_C\_\_\_\_\_\_dx+{鈭�}_C\_\_\_\_\_\_dy \\ =鈭�{鈭�}_R\left(\_\_\_\_\_\_-\_\_\_\_\_\_\right)dA. \end{gathered}$$

Make a chart of all the new notation in this section, including what each item means in terms you already understand.

Make a chart of all the new notation, definitions, and theorems in this section, including what each new thing means in terms you already understand.

What are the inputs of a vector field in ${\mathrm{鈩�}}^3$?