91影视

Evaluate the line integral of the given function over the specified curve.

${鈭�}_{C}F\left(x,y\right)路dr$where $F\left(x,y\right)=\frac{x}{x^2+y^2}i-\frac{y}{x^2+y^2}j$ and C is unit circle centered at origin.

Consider the vector field $\mathrm{F}(x,y,z)=(yz,xz,xy)$. Find a vector field $\mathrm{G}(x,y,z)$ with the property that, for all points in role="math" localid="1650383268941" ${\mathrm{鈩�}}^3,\mathrm{G}(x,y,z)=2\mathrm{F}(x,y,z)$.

Give an example of a non-conservative vector field whose divergence is never equal to zero in ${\mathrm{鈩�}}^3$ .

Compute n for the surface S in Exercise 12.

Write ${鈭�}_{\mathrm{C}}\mathbf{F}(x,y)路\mathrm{d}r$ explicitly as an integral of t, where $\mathbf{F}(x,y)=(x^{2}y,x鈭�y)$ and $\mathbf{r}\left(t\right)=(2t,e^t)$ for role="math" localid="1650821016448" $a鈮�t鈮�b$.

Why is $dA$in Green鈥檚 Theorem replaced by $dS$in Stokes鈥� Theorem?

$$\begin{gathered} \mathrm{Use}\mathrm{the}\mathrm{vector}\mathrm{field}\mathrm{F}(\mathrm{x},\mathrm{y})={\mathrm{x}}^2{\mathrm{e}}^{\mathrm{y}}\mathrm{i}+\cos \left(\mathrm{x}\right)\sin \left(\mathrm{y}\right)\mathrm{j}\mathrm{and}\mathrm{Green}鈥�\mathrm{s}\mathrm{Theorem}\mathrm{to}\mathrm{write} \\ \mathrm{the}\mathrm{line}\mathrm{integral}\mathrm{of}\mathrm{F}(\mathrm{x},\mathrm{y})\mathrm{about}\mathrm{the}\mathrm{unit}\mathrm{circle},\mathrm{traversed}\mathrm{counterclockwise},\mathrm{as}\mathrm{a} \\ \mathrm{double}\mathrm{integral}.\mathrm{Do}\mathrm{not}\mathrm{evaluate}\mathrm{the}\mathrm{integral}. \end{gathered}$$

Write${鈭�}_C\mathbf{F}(x,y,z)路d\mathbf{r}$explicitly as an integral of$t$, where$\mathrm{F}(x,y,z)=\{zy-x,xz-y,xy-z鉄�$and$\mathbf{r}\left(t\right)=鉄�t,\cos t,\sin t鉄�$for$a鈮�t鈮�b$

Evaluate the line integral of the given function over the specified curve.

${鈭�}_{c}F\left(x,y,z\right)路dr$where $F\left(x,y,z\right)=yzi+xzj+xyk$and C is the curve parametrized by$x=t^2,y=t^{-2},z=t$for$1鈮�t鈮�3$.

Write ${鈭�}_{\mathrm{C}}\mathbf{F}(x,y,z)路\mathrm{d}r$ explicitly as an integral of t, where $\mathbf{F}(x,y,z)=(zy鈭�x,xz鈭�y,xy鈭�z)$ and role="math" localid="1650820575074" $\mathbf{r}\left(t\right)=(t,\cos t,\sin t)$ for $a鈮�t鈮�b$.