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Define the vector field F

$⟨2 x z\+y^2\,2 x y-z^3\,x^2-3 y z^2⟩$ = $\stackrel{\text{to Vector Field}}{\to }$ $\stackrel{\text{assign to a name}}{\to }$$F$$$

Obtain a scalar potential for F

$\mathbf{F}$ = $\stackrel{\text{scalar potential}}{\to }$$-y{z}^3\+x^{2}z\+x{y}^2$$$

Table 9.7.8(a)   Solution by the ScalarPotential command accessed through the Context Panel

Define the components of F as the functions $f\left(x\,y\,z\right)$, $g\left(x\,y\,z\right)$,  and $h\left(x\,y\,z\right)$ 

Implement Recipe 1, Table 9.7.4

${\int }_a^{x}f\left(t\,b\,c\right) \ⅆt\+{\int }_b^{y}g\left(x\,t\,c\right) \ⅆt\+{\int }_c^{z}h\left(x\,y\,t\right) \ⅆt$

$b^2\left(x-a\right)\+c\left(-a^2\+x^2\right)-c^3\left(y-b\right)\+x\left(-b^2\+y^2\right)-y\left(-c^3\+z^3\right)\+x^2\left(z-c\right)$

$\stackrel{\text{simplify}}{\=}$

$b{c}^3-y{z}^3-a^{2}c-a{b}^2\+x^{2}z\+x{y}^2$

$\stackrel{\text{evaluate at point}}{\to }$

$-y{z}^3\+x^{2}z\+x{y}^2$

Table 9.7.8(b)   Implementation of Recipe 1, Table 9.7.4

Initialize

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

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

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

Use the ScalarPotential command to obtain a scalar potential for F

$\mathrm{ScalarPotential}\left(\mathbf{F}\right)$$$ = $-y{z}^3\+x^{2}z\+x{y}^2$$$

Table 9.7.8(c)   Application of the ScalarPotential command

Use the unapply command to define the components of F as the functions $f\left(x\,y\,z\right)$, $g\left(x\,y\,z\right)$,  and $h\left(x\,y\,z\right)$ 

Use the Int command to write the integrals of Recipe 1, Table 9.7.4

$\mathrm{Int}\left(f\left(t\,b\,c\right)\,t\=a..x\right)\+\mathrm{Int}\left(g\left(x\,t\,c\right)\,t\=b..y\right)\+\mathrm{Int}\left(h\left(x\,y\,t\right)\,t\=c..z\right)$

${\∫}_a^x\left(b^2\+2ct\right)\ⅆt\+{\∫}_b^y\left(-c^3\+2tx\right)\ⅆt\+{\∫}_c^z\left(-3t^{2}y\+x^2\right)\ⅆt$

Access the top-level int command by prefixing "colon-dash" to int

$u≔:-\mathrm{int}\left(f\left(t\,b\,c\right)\,t\=a..x\right)\+:-\mathrm{int}\left(g\left(x\,t\,c\right)\,t\=b..y\right)\+:-\mathrm{int}\left(h\left(x\,y\,t\right)\,t\=c..z\right)$

$b^2\left(x-a\right)\+c\left(-a^2\+x^2\right)-c^3\left(y-b\right)\+x\left(-b^2\+y^2\right)-y\left(-c^3\+z^3\right)\+x^2\left(z-c\right)$

$\mathrm{simplify}\left(\mathrm{eval}\left(u\,\left[a\=0\,b\=0\,c\=0\right]\right)\right)$

$-y{z}^3\+x^{2}z\+x{y}^2$

Table 9.7.8(d)   Implementation of Recipe 1, Table 9.7.4