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Define the vector field F

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

Form and evaluate the line integral of F along $C_1$ 

$\mathbf{F}$ = $\stackrel{\text{line integral}}{\to }$${\∫}_0^1\left(-4t^3\+6t^2\right)\ⅆt$$\=$$1$

Form and evaluate the line integral of F along $C_2$ 

$\mathbf{F}$ = $\stackrel{\text{line integral}}{\to }$${\∫}_0^{1}0\ⅆt\+{\∫}_0^{1}2t\ⅆt\+{\∫}_0^1\left(-3t^2\+1\right)\ⅆt$$\=$$1$

Table 9.7.5(a)   Solution via the LineInt command accessed through the Context Panel

Form a list of nodes for the polygonal line $C_2$

$\left[⟨0\,0\,0⟩\,⟨1\,0\,0⟩\,⟨1\,1\,0 ⟩\,⟨1\,1\,1⟩\right]$$\stackrel{\text{assign to a name}}{\to }$$P$

Form a position-vector representation of each segment of $C_2$ 

${\mathbf{P}}_1\+t\cdot \left({\mathbf{P}}_2-{\mathbf{P}}_1\right)$ = $\stackrel{\text{assign to a name}}{\to }$$r_1$

${\mathbf{P}}_2\+t\cdot \left({\mathbf{P}}_3-{\mathbf{P}}_2\right)$ = $\stackrel{\text{assign to a name}}{\to }$$r_2$$$

${\mathbf{P}}_3\+t\cdot \left({\mathbf{P}}_4-{\mathbf{P}}_3\right)$ = $\stackrel{\text{assign to a name}}{\to }$$r_3$$$

Obtain a parametric representation of each line segment in $C_2$

$⟨x\,y\,z⟩\,{\mathbf{r}}_1$ = $\stackrel{\text{equate}}{\to }$$\left[x\=t\,y\=0\,z\=0\right]$$\stackrel{\text{assign to a name}}{\to }$$s_1$$$

$⟨x\,y\,z⟩\,{\mathbf{r}}_2$ = $\stackrel{\text{equate}}{\to }$$\left[x\=1\,y\=t\,z\=0\right]$$\stackrel{\text{assign to a name}}{\to }$$s_2$$$

$⟨x\,y\,z⟩\,{\mathbf{r}}_3$ = $\stackrel{\text{equate}}{\to }$$\left[x\=1\,y\=1\,z\=t\right]$$\stackrel{\text{assign to a name}}{\to }$$s_3$$$

Evaluate $\mathbf{F}·\mathbf{dr}$ on each segment of $C_2$ 

Form and evaluate line integrals on each segment of $C_2$ 

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

Table 9.7.5(b)   Line integral on $C_2$ from first principles

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

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

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

Form and evaluate the line integral of F along $C_1$ 

$\mathrm{LineInt}\left(\mathbf{F}\,\mathrm{Line}\left(⟨0\,0\,0⟩\,⟨1\,1\,1⟩\right)\,\mathrm{output}\=\mathrm{integral}\right)$ = ${\∫}_0^1\left(-4t^3\+6t^2\right)\ⅆt$$$

$\mathrm{LineInt}\left(\mathbf{F}\,\mathrm{Line}\left(⟨0\,0\,0⟩\,⟨1\,1\,1⟩\right)\right)$ = $1$$$

Form and evaluate the line integral of F along $C_2$ 

$\mathrm{LineInt}\left(\mathbf{F}\,\mathrm{LineSegments}\left(⟨0\,0\,0⟩\,⟨1\,0\,0⟩\,⟨1\,1\,0⟩\,⟨1\,1\,1⟩\right)\,\mathrm{output}\=\mathrm{integral}\right)$

${\∫}_0^{1}0\ⅆt\+{\∫}_0^{1}2t\ⅆt\+{\∫}_0^1\left(-3t^2\+1\right)\ⅆt$

$\mathrm{LineInt}\left(\mathbf{F}\,\mathrm{LineSegments}\left(⟨0\,0\,0⟩\,⟨1\,0\,0⟩\,⟨1\,1\,0⟩\,⟨1\,1\,1⟩\right)\right)$ = $1$$$

Table 9.7.5(c)   Solution via the LineInt command

Define the nodes on $C_2$ 

$P≔\left[⟨0\,0\,0⟩\,⟨1\,0\,0⟩\,⟨1\,1\,0 ⟩\,⟨1\,1\,1⟩\right]\:$

Form a position-vector representation of each segment of $C_2$ 

Use the Equate command to obtain a parametric representation of each line segment in $C_2$ 

Evaluate $\mathbf{F}·\mathbf{dr}$ on each segment of $C_2$ 

$\mathrm{eval}\left(\mathrm{DotProduct}\left(\mathbf{F}\,\mathrm{diff}\left({\mathbf{r}}_1\,t\right)\right)\,s_1\right)$ = $0$$$

$\mathrm{eval}\left(\mathrm{DotProduct}\left(\mathbf{F}\,\mathrm{diff}\left({\mathbf{r}}_2\,t\right)\right)\,s_2\right)$ = $2t$$$

$\mathrm{eval}\left(\mathrm{DotProduct}\left(\mathbf{F}\,\mathrm{diff}\left({\mathbf{r}}_3\,t\right)\right)\,s_3\right)$ = $-3t^2\+1$$$

Use the Int and top-level int commands to form and evaluate line integrals on each segment of $C_2$ 

$$\begin{gathered} \mathrm{Int}\left(0\,t\=0..1\right)\+\mathrm{Int}\left(2 t\,t\=0..1\right)\+\mathrm{Int}\left(1-3 t^2\,t\=0..1\right)\= \\ :-\mathrm{int}\left(0\,t\=0..1\right)\+:-\mathrm{int}\left(2 t\,t\=0..1\right)\+:-\mathrm{int}\left(1-3 t^2\,t\=0..1\right) \end{gathered}$$

${\∫}_0^{1}0\ⅆt\+{\∫}_0^{1}2t\ⅆt\+{\∫}_0^1\left(-3t^2\+1\right)\ⅆt\=1$

Table 9.7.5(d)   Line integral along $C_2$ implemented from first principles