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Define the vector field F

$⟨2 x y-y^3\,x^2-3 x y^2⟩$ = $\left[\begin{array}{c}-y^3+2xy\\ -3x{y}^2+x^2\end{array}\right]$$\stackrel{\text{to Vector Field}}{\to }$$\left[\begin{array}{c}-y^3+2xy\\ -3x{y}^2+x^2\end{array}\right]$$\stackrel{\text{assign to a name}}{\to }$$F$$$

Form and evaluate the requisite line integral

$\mathbf{F}$ = $\stackrel{\text{line integral}}{\to }$${\int }_0^{2\mathrm{\pi }}\left(-r\left(-{\left(b+r\sin \left(t\right)\right)}^3+2\left(a+r\cos \left(t\right)\right)\left(b+r\sin \left(t\right)\right)\right)\sin \left(t\right)+r\left(-3\left(a+r\cos \left(t\right)\right){\left(b+r\sin \left(t\right)\right)}^2+{\left(a+r\cos \left(t\right)\right)}^2\right)\cos \left(t\right)\right)\ⅆt$$$

$\mathbf{F}$ = $\stackrel{\text{line integral}}{\to }$$0$$$$$

Table 9.7.1(a)   Context Panel access to the LineInt command

Parametric definition of the general circle in the plane

$a\+r \cos \left(t\right)$$\stackrel{\text{assign to a name}}{\to }$$X$

$b\+r \sin \left(t\right)$$\stackrel{\text{assign to a name}}{\to }$$Y$

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

${\int }_0^{2 \mathrm{\π}}\left(\genfrac{}{}{0ex}{}{{\mathbf{F}}_1}{\phantom{x\=a}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{f(x)}}}{x\=X\,y\=Y}\right)\cdot \left(\frac{\ⅆ}{\ⅆ t} X\right) \mathit{\ⅆ}t\+{\int }_0^{2 \mathrm{\π}}\left(\genfrac{}{}{0ex}{}{{\mathbf{F}}_2}{\phantom{x\=a}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{f(x)}}}{x\=X\,y\=Y}\right)\cdot \left(\frac{\ⅆ}{\ⅆ t} Y\right) \mathit{\ⅆ}t$ = $0$$$

Table 9.7.1(b)   Solution from first principles

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

$\mathbf{F}≔\mathrm{VectorField}\left(⟨2 x y-y^3\,x^2-3 x y^2⟩\right)\:$

Form and evaluate the requisite line integral with the LineInt command

$\mathrm{LineInt}\left(\mathbf{F}\,\mathrm{Circle}\left(⟨a\,b⟩\,r\right)\,\mathrm{output}\=\mathrm{integral}\right)$

${\∫}_0^{2\mathrm{\π}}\left(-r\left(-{\left(b\+r\sin \left(t\right)\right)}^3\+2\left(a\+r\cos \left(t\right)\right)\left(b\+r\sin \left(t\right)\right)\right)\sin \left(t\right)\+r\left(-3\left(a\+r\cos \left(t\right)\right){\left(b\+r\sin \left(t\right)\right)}^2\+{\left(a\+r\cos \left(t\right)\right)}^2\right)\cos \left(t\right)\right)\ⅆt$

$\mathrm{LineInt}\left(\mathbf{F}\,\mathrm{Circle}\left(⟨a\,b⟩\,r\right)\right)$ = $0$$$

Table 9.7.1(c)   Solution via the LineInt command

$X\,Y≔a\+r \cos \left(t\right)\,b\+r \sin \left(t\right)\:$

$$\begin{gathered} \mathrm{Int}\left(\mathrm{eval}\left(F_1\,\left[x\=X\,y\=Y\right]\right)\cdot \mathrm{diff}\left(X\,t\right)\,t\=0..2 \mathrm{\π}\right)\+ \\ \mathrm{Int}\left(\mathrm{eval}\left(F_2\,\left[x\=X\,y\=Y\right]\right)\cdot \mathrm{diff}\left(Y\,t\right)\,t\=0..2 \mathrm{\π}\right) \end{gathered}$$

${\∫}_0^{2\mathrm{\π}}\left(-\left(-{\left(b\+r\sin \left(t\right)\right)}^3\+2\left(a\+r\cos \left(t\right)\right)\left(b\+r\sin \left(t\right)\right)\right)r\sin \left(t\right)\right)\ⅆt\+{\∫}_0^{2\mathrm{\π}}\left(-3\left(a\+r\cos \left(t\right)\right){\left(b\+r\sin \left(t\right)\right)}^2\+{\left(a\+r\cos \left(t\right)\right)}^2\right)r\cos \left(t\right)\ⅆt$

$$\begin{gathered} :-\mathrm{int}\left(\mathrm{eval}\left(F_1\,\left[x\=X\,y\=Y\right]\right)\cdot \mathrm{diff}\left(X\,t\right)\,t\=0..2 \mathrm{\π}\right)\+ \\ :-\mathrm{int}\left(\mathrm{eval}\left(F_2\,\left[x\=X\,y\=Y\right]\right)\cdot \mathrm{diff}\left(Y\,t\right)\,t\=0..2 \mathrm{\π}\right) \end{gathered}$$

$0$

Table 9.7.1(d)   Solution from first principles