91影视

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.

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

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.

$\mathbf{F}(x,y,z)=y{z}^{xy}\ln z\mathbf{i}+x{z}^{xy}\ln z\mathbf{j}+\frac{z^{xy}}{z}\mathbf{k}$, with C any curve from $(0,0,1)\mathrm{to}(2,16,3)$.

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

41. , and $S$ is the surface of the rectangular solid with vertices $(1,1,1),(1,5,1),(1,1,4)$, $(1,5,4),(7,1,1),(7,5,1),(7,1,4)$, and $(7,5,4)$.

Consider once again the notion of the rotation of a vector field. If a vector field $F(x,y,z)$has curl role="math" localid="1650703855457" $F=0$at a point $P$, then the field is said to be irrotational at that point. Show that the fields in Exercise are irrotational at the given points.

$F(x,y,z)=鈭�f(x,y,z)$, where $f(x,y,z)$ is a function defined on ${\mathrm{鈩�}}^3$ and has continuous first and second partial derivatives, and where $P=(x_0,y_0,z_0)$.

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.

$$F(x, y,z) = (cos(zy) 鈭� sin(zy))i 鈭� (xz sin(zy) + xz cos(zy))j 鈭� (xy sin(zy) + xycos(zy)) k$$, with $$C$$ any curve from $$(5,\frac{\pi}{2}, \frac{\pi}{3})$$ to $$(2,0, 2\pi) $$.

$\mathbf{F}(x,y,z)=鈭�y\mathbf{i}+x\mathbf{j}鈭�e^{yz}\mathbf{k}$, where S is the cylinder with equation $x^2+y^2=9$from $z=2,4$, with n pointing outwards.

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

42. $\mathbf{F}(x,y,z)=2xz\mathbf{i}+4xy\mathbf{j}-8zy\mathbf{k}$, and $S$ is the surface determined by $y=4-x^2,y鈮�0,0鈮�z鈮�5$.

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.

$\mathbf{F}(x,y)=\left({\tan }^{鈭�1}y,\frac{x}{1+y^2}\right)$

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.

$\mathbf{F}(x,y,z)=z\sin x\mathbf{i}+x\ln (y+4)\mathbf{j}+y\mathbf{k}$, with C the unit circle traversed counterclockwise starting at the point (1, 0).

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.

$\mathbf{G}(x,y)=(2x+y\cos (xy),x\cos (xy)鈭�1)$