6.5 Divergence and curl  (Page 7/9)

If the coordinate functions of $\text{F}:{ℝ}^3\to {ℝ}^3$ have continuous second partial derivatives, then $\text{curl}(\text{div}(\text{F}))$ equals zero.

$\nabla ·\left(x\text{i}+y\text{j}+z\text{k}\right)=1.$

All vector fields of the form $\text{F}\left(x,y,z\right)=f\left(x\right)\text{i}+g\left(y\right)\text{j}+h\left(z\right)\text{k}$ are conservative.

If $\text{curl}\text{F}=0,$ then F is conservative.

If F is a constant vector field then $\text{div}\text{F}=0.$

If F is a constant vector field then $\text{curl}\text{F}=0.$

$\text{F}\left(x,y,z\right)=x{y}^2{z}^4\text{i}+\left(2x^{2}y+z\right)\text{j}+y^3{z}^2\text{k}$

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

$\text{F}\left(x,y,z\right)=3xy{z}^2\text{i}+y^2\text{sin}z\text{j}+x{e}^{2z}\text{k}$

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

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

$\text{F}(x,y,z)=(x-y)\text{i}+(y-z)\text{j}+(z-x)\text{k}$

$\text{F}(x,y,z)=xyz\text{i}+x^2{y}^2{z}^2\text{j}+y^2{z}^3\text{k}$

$\text{F}(x,y,z)=xy\text{i}+yz\text{j}+xz\text{k}$

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

$\text{F}(x,y,z)=ax\text{i}+by\text{j}+c\text{k}$ for constants a , b , c

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

$\text{F}(x,y,z)=3xy{z}^2\text{i}+y^2\text{sin}z\text{j}+x{e}^{2z}\text{k}$

$\text{F}(x,y)=(\text{sin}x)\text{i}+(\text{cos}y)\text{j}$

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

$\text{F}(x,y,z)=(x-y)\text{i}+(y-z)\text{j}+\left(z-x\right)\text{k}$

$\text{F}(x,y)=\frac{x}{\sqrt{x^2+y^2}}\text{i}+\frac{y}{\sqrt{x^2+y^2}}\text{j}$

$\text{F}(x,y)=x\text{i}-y\text{j}$

$\text{F}(x,y,z)=ax\text{i}+by\text{j}+c\text{k}$ for constants a , b , c

$\text{F}(x,y,z)=xyz\text{i}+x^2{y}^2{z}^2\text{j}+y^2{z}^3\text{k}$

$\text{F}(x,y,z)=xy\text{i}+yz\text{j}+xz\text{k}$

$u(x,y,z)=e^{\text{−}x}(\text{cos}y-\text{sin}y)$

$w(x,y,z)={\left(x^2+y^2+z^2\right)}^{\text{−}1\text{/}2}$

If $\text{F}(x,y,z)=2\text{i}+2x\text{j}+3y\text{k}$ and $\text{G}(x,y,z)=x\text{i}-y\text{j}+z\text{k},$ find $\text{curl}(\text{F}\times \text{G}).$

If $\text{F}(x,y,z)=2\text{i}+2x\text{j}+3y\text{k}$ and $\text{G}(x,y,z)=x\text{i}-y\text{j}+z\text{k},$ find $\text{div}(\text{F}\times \text{G}).$

Find $\text{div}\text{F},$ given that $\text{F}=\text{∇}f,$ where $f(x,y,z)=x{y}^3{z}^2.$

Find the divergence of F for vector field $\text{F}(x,y,z)=\left(y^2+z^2\right)\left(x+y\right)\text{i}+\left(z^2+x^2\right)\left(y+z\right)\text{j}+\left(x^2+y^2\right)\left(z+x\right)\text{k}.$

Find the divergence of F for vector field $\text{F}(x,y,z)=f_1(y,z)\text{i}+f_2(x,z)\text{j}+f_3(x,y)\text{k}.$

Find the $\text{curl}\text{r}.$

Find the $\text{curl}\frac{\text{r}}{r}.$

Find the $\text{curl}\frac{\text{r}}{r^3}.$

Let $\text{F}(x,y)=\frac{\text{−}y\text{i}+x\text{j}}{x^2+y^2},$ where F is defined on $\left\{(x,y)\in ℝ|(x,y)\ne (0,0)\right\}.$ Find $\text{curl}\text{F}.$

[T] $\text{F}(x,y,z)=\text{arctan}\left(\frac{x}{y}\right)\text{i}+\text{ln}\sqrt{x^2+y^2}\text{j}+\text{k}$

[T] $\text{F}(x,y,z)=\text{sin}(x-y)\text{i}+\text{sin}(y-z)\text{j}+\text{sin}(z-x)\text{k}$

$\text{F}(x,y,z)=\text{i}+\text{j}+\text{k}$ at $(2,\mathrm{-1},3)$

$\text{F}(x,y,z)=xyz\text{i}+y\text{j}+z\text{k}$ at $(1,2,3)$

$\text{F}(x,y,z)=e^{\text{−}xy}\text{i}+e^{xz}\text{j}+e^{yz}\text{k}$ at $(3,2,0)$

$\text{F}(x,y,z)=xyz\text{i}+y\text{j}+z\text{k}$ at (1, 2, 1)

$\text{F}(x,y,z)=e^x\text{sin}y\text{i}-e^x\text{cos}y\text{j}$ at (0, 0, 3)

$\text{F}(x,y,z)=\text{i}+\text{j}+\text{k}$ at $(2,\mathrm{-1},3)$

$\text{F}(x,y,z)=xyz\text{i}+y\text{j}+x\text{k}$ at $(1,2,3)$

$\text{F}(x,y,z)=e^{\text{−}xy}\text{i}+e^{xz}\text{j}+e^{yz}\text{k}$ at (3, 2, 0)

$\text{F}(x,y,z)=xyz\text{i}+y\text{j}+z\text{k}$ at (1, 2, 1)

$\text{F}(x,y,z)=e^x\text{sin}y\text{i}-e^x\text{cos}y\text{j}$ at (0, 0, 3)

Let $\text{F}(x,y,z)=\left(3x^{2}y+az\right)\text{i}+x^3\text{j}+\left(3x+3z^2\right)\text{k}.$ For what value of a is F conservative?

Given vector field $\text{F}(x,y)=\frac{1}{x^2+y^2}\left(\text{−}y,x\right)$ on domain $D=\frac{{ℝ}^2}{\left\{\left(0,0\right)\right\}}=\left\{(x,y)\in {ℝ}^2|(x,y)\ne (0,0)\right\},$ is F conservative?

Given vector field $\text{F}(x,y)=\frac{1}{x^2+y^2}\left(x,y\right)$ on domain $D=\frac{{ℝ}^2}{\left\{\left(0,0\right)\right\}},$ is F conservative?

Find the work done by force field $\text{F}(x,y)=e^{\text{−}y}\text{i}-x{e}^{\text{−}y}\text{j}$ in moving an object from P (0, 1) to Q (2, 0). Is the force field conservative?

Compute divergence $\text{F}=\left(\text{sinh}x\right)\text{i}+\left(\text{cosh}y\right)\text{j}-xyz\text{k}.$

Compute curl $\text{F}=\left(\text{sinh}x\right)\text{i}+\left(\text{cosh}y\right)\text{j}-xyz\text{k}.$

Express F in terms of i , j , and k vectors.

Find $\text{div}\text{F}.$

Find $\text{curl}\text{F}$

Does $\text{F}+\text{G}$ necessarily have zero divergence?

Does $\text{F}\times \text{G}$ necessarily have zero divergence?

Compute the heat flow vector field.

Compute the divergence.

[T] Consider rotational velocity field $\text{v}=⟨0,10z,\mathrm{-10}y⟩.$ If a paddlewheel is placed in plane $x+y+z=1$ with its axis normal to this plane, using a computer algebra system, calculate how fast the paddlewheel spins in revolutions per unit time.