91影视

In Exercises 41-44, find the fluxes of the vector fields through the given surfaces in the direction of the outwards-pointing normal vector.

43. $\mathbf{F}(x,y,z)=鉄�y\cos z,3y,\sin \left(xy\right))$, and $S$is the surface of the pyramid with the square base in the $xy$-plane and with vertices $(1,1,0),(-1,1,0),(1,-1,0),(-1,-1,0)$and apex $(0,0,4)$.

Find${鈭�}_S鈥�1dS$, where S is the portion of the surface determined by$z=x^2鈭�\sqrt{3}y$that lies above the region in the xy-plane bounded by the x-axis and the lines with equations$$\begin{gathered} y=2x, \\ x=3 \end{gathered}$$.

Find

$\begin{array}{r}{鈭�}_S鈥�\mathbf{F}(x,y,z)鈰�\mathbf{n}dS\text{if}\\ \mathbf{F}(x,y,z)=\frac{\ln 鈦�\left(x^2+y^2+1\right)}{z+3}\mathbf{i}+\frac{y}{y+1}\mathbf{j}+e^{z^2}\mathbf{k}\end{array}$

Where S is the portion of the sphere with radius 2, centered at the origin, and that lies below the plane with equation $z=-\sqrt{2}$, with n pointing outwards.

Determine whether or not each of the vector fields in Exercises 41鈥�48 is conservative. If the vector field is conservative, find a potential function for the field.

$\mathbf{G}(x,y)=y{x}^2\mathbf{i}+e^y\mathbf{j}$

In Exercises 41鈥�44, find the fluxes of the vector fields through the given surfaces in the direction of the outwards-pointing normal vector.

$$F(x,y,z)= \langle x^{2}y^{2}z^{2}, z-x-y,\frac{y}{z+1} \rangle$$, and $$S$$ is the surface of the region bounded by $$z = x^{2}$$, $$x = 0$$, and $$4x +2z = 4$$.

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.

$\mathbf{F}(x,y,z)=(z^3+1)\mathbf{i}+x\cos z\mathbf{j}+xy{e}^{xyz}\mathbf{k}$, with C the curve parameterized by $\mathbf{r}\left(t\right)=t^2\mathbf{i}+t^3\mathbf{j}鈭�t\mathbf{k}$ for $0鈮�t鈮�4$.

Determine whether or not each of the vector fields in Exercises 41鈥�48 is conservative. If the vector field is conservative, find a potential function for the field.

$\mathbf{F}(x,y,z)=(y{e}^{2z}+1,x{e}^{2z},2xy{e}^{2z})$

Evaluate the line integrals in Exercises 45鈥�50.

${鈭�}_{\mathrm{C}}\mathbf{F}(x,y)路d\mathbf{r},\mathrm{where}\mathbf{F}(x,y)=鈭�y\mathbf{i}+x\mathbf{j}$ and C is the spiral $x=t\cos t,y=t\sin t,\mathrm{for}\mathrm{蟺}鈮�t鈮�2\mathrm{蟺}$.

The current through a certain region of the San Juan Islands in Washington State is given by

. Consider a disk $R$of radius $1$mile centered on this region. Denote the boundary of the disk by $鈭�R$.

(a) Compute ${鈭�}_R鈭�脳\mathbf{F}\mathbf{脳}\mathbf{n}dA$.

(b) Show that,

${鈭�}_{鈭�R}\mathbf{F}\mathbf{脳}d\mathbf{r}={鈭�}_0^{2蟺}\left(1.152-0.8{\cos }^2胃\right)\sin 胃d胃=0$

Conclude that Stokes鈥� Theorem is valid for the current in this region of the San Juan Islands.

(c) What do the integrals from Stokes鈥� Theorem tell us about this region of the San Juan Islands?

45. The current through a certain region of the San Juan Islands in Washington State is given by $F=(0,1.152)-0.8x^2$. Consider a disk $R$ of radius 1 centered on this region. Denote the circle that comprises the boundary of the disk by $鈭�R$.

(a) Compute ${鈭�}_R鈭�路\mathbf{F}dA$.

(b) Show that ${鈭�}_{aR}\mathbf{F}路\mathbf{n}ds=0$. Conclude that the Divergence Theorem is valid for the current in this region of the San Juan Islands.

(c) What do the values of the integrals from the Divergence Theorem tell us about this region of the San Juan Islands?