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Loading Student:-VectorCalculus

Define the vector field F

$⟨x\,y\,z⟩$ = $\stackrel{\text{to Vector Field}}{\to }$ $\stackrel{\text{assign to a name}}{\to }$$F$$$

Obtain $\nabla ·\mathbf{F}$, the divergence of F

$\nabla ·\mathbf{F}$ = $3$$$

Use cylindrical coordinates to obtain the volume integral of the divergence of F

Loading Student:-MultivariateCalculus

$3$$\stackrel{\text{MultiInt}}{\to }$${\∫}_0^{2\mathrm{\π}}{\∫}_0^1{\∫}_0^{1}3r\ⅆz\ⅆr\ⅆ\mathrm{\θ}$$\=$$3\mathrm{\π}$$$

Use a task template to obtain the flux of F through the cylinder

Tools≻Tasks≻Browse:
Calculus - Vector≻Integration≻Flux≻3-D≻Through a Parametric Surface

Use a task template to obtain the flux of F through the top of the cylinder

Tools≻Tasks≻Browse:
Calculus - Vector≻Integration≻Flux≻3-D≻Through a Surface Defined over a Disk

Use the same task template to obtain the flux of F through the bottom of the cylinder

Tools≻Tasks≻Browse:
Calculus - Vector≻Integration≻Flux≻3-D≻Through a Surface Defined over a Disk

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$\mathrm{with}\left(\mathrm{Student}:-\mathrm{VectorCalculus}\right)\:$

$\mathrm{BasisFormat}\left(\mathrm{false}\right)\:$

$\mathbf{F}≔\mathrm{VectorField}\left(⟨x\,y\,z⟩\right)\:$

Obtain $\nabla ·\mathbf{F}$, the divergence of F

$\mathrm{Divergence}\left(\mathbf{F}\right)$ = $3$$$

Use the int command to integrate the divergence of F over $R$

$\mathrm{int}\left(3\,\left[x\,y\,z\right]\=\mathrm{Region}\left(-1..1\,-\sqrt{1-x^2}..\sqrt{1-x^2}\,0..1\right)\,\mathrm{output}\=\mathrm{integral}\right)$

${\∫}_{-1}^1{\∫}_{-\sqrt{-x^2\+1}}^{\sqrt{-x^2\+1}}{\∫}_0^{1}3\ⅆz\ⅆy\ⅆx$

$\mathrm{int}\left(3\,\left[x\,y\,z\right]\=\mathrm{Region}\left(-1..1\,-\sqrt{1-x^2}..\sqrt{1-x^2}\,0..1\right)\right)$ = $3\mathrm{\π}$$$

Use the Flux command to obtain the flux of F through the cylinder

$S≔\mathrm{Vector}\left(⟨1\,\mathrm{\θ}\,z⟩\,\mathrm{cylindrical}\right)\:$

$$\begin{gathered} \mathrm{Flux}\left(\mathbf{F}\,\mathrm{Surface}\left(S\,\mathrm{\θ}\=0..2 \mathrm{\π}\,z\=0..1\right)\,\mathrm{output}\=\mathrm{integral}\right)\+ \\ \mathrm{Flux}\left(\mathbf{F}\,\mathrm{Surface}\left(⟨x\,y\,1⟩\,\left[x\,y\right]\=\mathrm{Circle}\left(⟨0\,0⟩\,1\,\left[r\,\mathrm{\θ}\right]\right)\right)\,\mathrm{output}\=\mathrm{integral}\right)\+ \\ \mathrm{Flux}\left(\mathbf{F}\,\mathrm{Surface}\left(⟨x\,y\,0⟩\,\left[x\,y\right]\=\mathrm{Circle}\left(⟨0\,0⟩\,1\,\left[r\,\mathrm{\θ}\right]\right)\right)\,\mathrm{output}\=\mathrm{integral}\right) \end{gathered}$$

${\∫}_0^{2\mathrm{\π}}{\∫}_0^1\left({\cos \left(\mathrm{\θ}\right)}^2\+{\sin \left(\mathrm{\θ}\right)}^2\right)\ⅆz\ⅆ\mathrm{\θ}\+{\∫}_0^1{\∫}_0^{2\mathrm{\π}}r\ⅆ\mathrm{\θ}\ⅆr\+{\∫}_0^1{\∫}_0^{2\mathrm{\π}}0\ⅆ\mathrm{\θ}\ⅆr$

$$\begin{gathered} \mathrm{Flux}\left(\mathbf{F}\,\mathrm{Surface}\left(S\,\mathrm{\theta }\=0..2 \mathrm{\pi }\,z\=0..1\right)\right)\+ \\ \mathrm{Flux}\left(\mathbf{F}\,\mathrm{Surface}\left(⟨x\,y\,1⟩\,\left[x\,y\right]\=\mathrm{Circle}\left(⟨0\,0⟩\,1\right)\right)\right)\+ \\ \mathrm{Flux}\left(\mathbf{F}\,\mathrm{Surface}\left(⟨x\,y\,0⟩\,\left[x\,y\right]\=\mathrm{Circle}\left(⟨0\,0⟩\,1\right)\right)\right) \end{gathered}$$

$3\mathrm{\π}$