$\nabla ·\mathbf{F}$

$\={\partial }_x\left(3 x^2-3 y^2\right)\+{\partial }_y\left(-6 x y\right)$

$\=6 x-6 x$

$\=0$

${\nabla }^{2}u\=u_{\mathrm{xx}}\+u_{\mathrm{yy}}$

$\=6 x-6 x$

$\=0$

Initialize

Loading Student:-VectorCalculus

Define the vector field F

$⟨3\left(x^2-y^2\right)\,-6 x y\,0⟩$ = $\stackrel{\text{to Vector Field}}{\to }$ $\stackrel{\text{assign to a name}}{\to }$$F$$$

Show $\nabla ·\mathbf{F}\=0$ and $\nabla \times \mathbf{F}\=\mathbf{0}$ 

$\nabla ·\mathbf{F}$ = $0$$$

$\nabla \times \mathbf{F}$ = $$

Obtain u, Maple's scalar potential for F

$\mathbf{F}$ = $\stackrel{\text{scalar potential}}{\to }$$x^3-3x{y}^2$$\stackrel{\text{assign to a name}}{\to }$$u$$$

Obtain Maple's vector potential for F

$\mathbf{F}$ = $\stackrel{\text{vector potential}}{\to }$ $\stackrel{\text{assign to a name}}{\to }$$A$$$

Table 9.7.12(a)   F is solenoidal and conservative with scalar and vector potentials

Obtain the Laplacian of $u$ via Typeset math notation

${\nabla }^{2}u$ = $0$$$

Alternately, obtain the Laplacian from first principles

$\frac{{\partial }^2}{{\partial }^{}{x}^2} u\+\frac{{\partial }^2}{{\partial }^{}{y}^2} u$ = $0$$$

Table 9.7.12(b)   Show $u$ is harmonic by showing its Laplacian vanishes

Initialize

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

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

$\mathbf{F}≔\mathrm{VectorField}\left(⟨3\left(x^2-y^2\right)\,-6 x y\,0⟩\right)\:$

Show $\nabla ·\mathbf{F}\=0$ and $\nabla \times \mathbf{F}\=\mathbf{0}$ 

$\mathrm{Divergence}\left(\mathbf{F}\right)$ = $0$$$

$\mathrm{Curl}\left(\mathbf{F}\right)$ = $$

Obtain Maple's scalar potential for F

$u≔\mathrm{ScalarPotential}\left(\mathbf{F}\right)$

$x^3-3x{y}^2$

Obtain Maple's vector potential for F

$\mathbf{A}≔\mathrm{VectorPotential}\left(\mathbf{F}\right)$

Table 9.7.11(c)   F is solenoidal and conservative; scalar and vector potentials for F 

Obtain the Laplacian of $u$ via the Laplacian command

$\mathrm{Laplacian}\left(u\right)$ = $0$$$

Alternatively, obtain the Laplacian of $u$ from first principles

$\mathrm{diff}\left(u\,x\,x\right)\+\mathrm{diff}\left(u\,y\,y\right)$ = $0$$$

Table 9.7.12(d)   Show $u$ is harmonic by showing its Laplacian vanishes