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Loading Student:-VectorCalculus

Define the Cartesian vector field F

$⟨x-2 y\,x^{2}y\,0⟩$ = $\stackrel{\text{to Vector Field}}{\to }$ $\stackrel{\text{assign to a name}}{\to }$$F$$$

Using polar coordinates, integrate the third component of $\nabla \times \mathbf{F}$ over the region $R$ 

$1\/\sqrt{{\cos }^2\left(\mathrm{\theta }\right)\+4 {\sin }^2\left(\mathrm{\theta }\right)}$$\stackrel{\text{assign to a name}}{\to }$$E$

$\genfrac{}{}{0ex}{}{{\left(\nabla \times \mathbf{F}\right)}_3}{\phantom{x\=a}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{f(x)}}}{x\=r \cos \left(\mathrm{\theta }\right)\,y\=r \sin \left(\mathrm{\theta }\right)}$ = $2r^2\cos \left(\mathrm{\θ}\right)\sin \left(\mathrm{\θ}\right)\+2$$\stackrel{\text{assign to a name}}{\to }$$q$

Loading Student:-MultivariateCalculus

$q$ = $2r^2\cos \left(\mathrm{\θ}\right)\sin \left(\mathrm{\θ}\right)\+2$$\stackrel{\text{MultiInt}}{\to }$${\∫}_0^{2\mathrm{\π}}{\∫}_0^E\left(2r^2\cos \left(\mathrm{\θ}\right)\sin \left(\mathrm{\θ}\right)\+2\right)r\ⅆr\ⅆ\mathrm{\θ}$$\=$$\mathrm{\π}$

Redefine F as a planar vector field

$⟨x-2 y\,x^{2}y⟩$ = $\stackrel{\text{to Vector Field}}{\to }$ $\stackrel{\text{assign to a name}}{\to }$$F$$$

Form and evaluate the line integral ${∳}_{C}f \mathrm{dx}\+g \mathrm{dy}$ 

$\mathbf{F}$ = $\stackrel{\text{line integral}}{\to }$${\∫}_0^{2\mathrm{\π}}\left(-\left(\cos \left(t\right)-\sin \left(t\right)\right)\sin \left(t\right)\+\frac{1}{4}{\cos \left(t\right)}^3\sin \left(t\right)\right)\ⅆt$$\=$$\mathrm{\π}$$$

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

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

$\mathbf{F}≔\mathrm{VectorField}\left(⟨x-2 y\,x^{2}y\,0⟩\right)\:$

Use the int command to integrate the third component of $\nabla \times \mathbf{F}$ over the region $R$ 

$\mathrm{int}\left(\mathrm{Curl}{\left(\mathbf{F}\right)}_3\,\left[x\,y\right]\=\mathrm{Ellipse}\left( x^2\+4 y^2\=1\,\left[r\,\mathrm{\θ}\right]\right)\,\mathrm{output}\=\mathrm{integral}\right)$

${\∫}_0^{2\mathrm{\π}}{\∫}_0^{\frac{1}{2}\sqrt{\frac{{\tan \left(\mathrm{\θ}\right)}^2\+1}{\frac{1}{4}\+{\tan \left(\mathrm{\θ}\right)}^2}}}\left(2r^2\cos \left(\mathrm{\θ}\right)\sin \left(\mathrm{\θ}\right)\+2\right)r\ⅆr\ⅆ\mathrm{\θ}$

$\mathrm{int}\left(\mathrm{Curl}{\left(\mathbf{F}\right)}_3\,\left[x\,y\right]\=\mathrm{Ellipse}\left( x^2\+4 y^2\=1\,\left[r\,\mathrm{\theta }\right]\right)\right)$ = $\mathrm{\π}$$$

Use the LineInt command to obtain ${∳}_{C}f \mathrm{dx}\+g \mathrm{dy}$ 

$\mathrm{LineInt}\left(\mathrm{VectorField}\left(⟨F_1\,F_2⟩\right)\,\mathrm{Ellipse}\left(x^2\+4 y^2\=1\right)\,\mathrm{output}\=\mathrm{integral}\right)$

${\∫}_0^{2\mathrm{\π}}\left(-\left(\cos \left(t\right)-\sin \left(t\right)\right)\sin \left(t\right)\+\frac{1}{4}{\cos \left(t\right)}^3\sin \left(t\right)\right)\ⅆt$

$\mathrm{LineInt}\left(\mathrm{VectorField}\left(⟨F_1\,F_2⟩\right)\,\mathrm{Ellipse}\left(x^2\+4 y^2\=1\right)\right)$ = $\mathrm{\π}$$$