${\int }_C\mathbf{F}·\mathbf{dr}$

$\={\int }_0^1\left(2\left(1\+2 t\right)\left(2\+t^3\right)\+{\left(2 t^2-t\+3\right)}^2\right) \mathrm{dt}$

$\+{\int }_0^1\left(2\left(1\+2 t\right)\left(2 t^2-t\+3\right)-{\left(2\+t^3\right)}^3\right) \mathrm{dt}$

$\+{\int }_0^1\left({\left(1\+2 t\right)}^2-3\left(2 t^2-t\+3\right){\left(2\+t^3\right)}^2\right) \mathrm{dt}$

$\=-{\int }_0^1\left(22t^{10}-10t^9\+27t^8\+96t^7-42t^6\+108t^5\+60t^4-48t^3\+39t^2\+14t-28\right)\mathrm{dt}$

$\=-20$

Initialize

Loading Student:-VectorCalculus

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$

Define the scalar potential as the function $u\left(x\,y\,z\right)$ 

$u\left(x\,y\,z\right)\=x^{2}z\+x{y}^2-y z^3$$\stackrel{\text{assign as function}}{\to }$$u$

Define the path $C$

Form and evaluate ${\int }_C\mathbf{F}·\mathbf{dr}$ 

Write the name F and press the Enter key.

Context Panel: Student Vector Calculus≻Line Integral
Complete the dialog as per

$\mathbf{F}$

$\stackrel{\text{line integral}}{\to }$

${\∫}_0^1\left(4\left(1\+2t\right)\left(t^3\+2\right)\+2{\left(2t^2-t\+3\right)}^2\+\left(-{\left(t^3\+2\right)}^3\+2\left(1\+2t\right)\left(2t^2-t\+3\right)\right)\left(4t-1\right)\+3\left(-3\left(2t^2-t\+3\right){\left(t^3\+2\right)}^2\+{\left(1\+2t\right)}^2\right)t^2\right)\ⅆt$

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

$-\left({\∫}_0^1\left(22t^{10}-10t^9\+27t^8\+96t^7-42t^6\+108t^5\+60t^4-48t^3\+39t^2\+14t-28\right)\ⅆt\right)$

$\=$

$-20$

Calculate $u\left(Q\right)-u\left(P\right)$ 

$u\left(3\,4\,3\right)-u\left(1\,3\,2\right)$ = $-20$

Table 9.7.10(a)   Verification that ${\int }_C\mathbf{F}·\mathbf{dr}\=u\left(Q\right)-u\left(P\right)$

Initialize

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

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

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

Define the scalar potential as the function $u\left(x\,y\,z\right)$ 

$u≔\left(x\,y\,z\right)\to x^{2}z\+x{y}^2-y z^3\:$

Define the path $C$

$X\,Y\,Z≔1\+2 t\,2 t^2-t\+3\,2\+t^3\:$

Use the LineInt and simplify commands to form and evaluate ${\int }_C\mathbf{F}·\mathbf{dr}$ 

$\mathrm{simplify}\left(\mathrm{LineInt}\left(F\,\mathrm{Path}\left(⟨X\,Y\,Z⟩\,t\=0..1\right)\,\mathrm{output}\=\mathrm{integral}\right)\right)$

$-\left({\∫}_0^1\left(22t^{10}-10t^9\+27t^8\+96t^7-42t^6\+108t^5\+60t^4-48t^3\+39t^2\+14t-28\right)\ⅆt\right)$

$\mathrm{LineInt}\left(F\,\mathrm{Path}\left(⟨X\,Y\,Z⟩\,t\=0..1\right)\right)$ = $-20$

Calculate $u\left(Q\right)-u\left(P\right)$ 

Use the eval command to obtain $P$ and $Q$ as lists, then remove the brackets by appending .

$u\left(\mathrm{eval}\left(\left[X\,Y\,Z\right]\,t\=1\right)\left[\right]\right)-u\left(\mathrm{eval}\left(\left[X\,Y\,Z\right]\,t\=0\right)\left[\right]\right)$ = $-20$

Table 9.7.10(b)   Verification that ${\int }_C\mathbf{F}·\mathbf{dr}\=u\left(Q\right)-u\left(P\right)$