${\mathit{e}}_{\mathbf{\ρ}}\=\left[\begin{array}{c}\cos \left(\mathrm{\theta }\right)\sin \(\mathrm{\φ}\)\\ \sin \left(\mathrm{\theta }\right)\sin \left(\mathrm{\phi }\right)\\ \cos \left(\mathrm{\phi }\right)\end{array}\right]$

${\mathit{e}}_{\mathbf{\φ}}\=\left[\begin{array}{c}\cos \left(\mathrm{\theta }\right)\cos \(\mathrm{\φ}\)\\ \sin \left(\mathrm{\theta }\right)\cos \left(\mathrm{\phi }\right)\\ -\sin \left(\mathrm{\phi }\right)\end{array}\right]$

${\mathit{e}}_{\mathbf{\θ}}\=\left[\begin{array}{c}-\sin \left(\mathrm{\theta }\right)\\ \cos \left(\mathrm{\theta }\right)\\ 0\end{array}\right]$

${\mathit{e}}_{\mathbf{\ρ}}\=\cos \left(\mathrm{\θ}\right)\sin \left(\mathrm{\φ}\right) \mathbf{i}\+\sin \left(\mathrm{\θ}\right)\sin \left(\mathrm{\φ}\right) \mathbf{j}\+\cos \left(\mathrm{\φ}\right) \mathbf{k}$

${\mathit{e}}_{\mathbf{\φ}}\=\cos \left(\mathrm{\θ}\right)\cos \left(\mathrm{\φ}\right) \mathbf{i}\+\sin \left(\mathrm{\θ}\right)\cos \left(\mathrm{\φ}\right) \mathbf{j}- \sin \left(\mathrm{\φ}\right) \mathbf{k}$

${\mathit{e}}_{\mathbf{\θ}}\= -\sin \left(\mathrm{\θ}\right) \mathbf{i}\+\cos \left(\mathrm{\θ}\right) \mathbf{j}$

$\mathbf{i}\=\cos \left(\mathrm{\θ}\right)\sin \left(\mathrm{\φ}\right) {\mathit{e}}_{\mathbf{\ρ}}\+\cos \left(\mathrm{\θ}\right)\cos \left(\mathrm{\φ}\right) {\mathit{e}}_{\mathbf{\φ}}-\sin \left(\mathrm{\θ}\right) {\mathit{e}}_{\mathbf{\θ}}$ 

$\mathbf{j}\=\sin \left(\mathrm{\θ}\right)\sin \left(\mathrm{\φ}\right) {\mathit{e}}_{\mathbf{\ρ}}\+\sin \left(\mathrm{\θ}\right)\cos \left(\mathrm{\φ}\right) {\mathit{e}}_{\mathbf{\φ}}\+\cos \left(\mathrm{\θ}\right) {\mathit{e}}_{\mathbf{\θ}}$ 

$\mathbf{k}\=\cos \left(\mathrm{\φ}\right) {\mathit{e}}_{\mathbf{\ρ}}-\sin \left(\mathrm{\φ}\right) {\mathit{e}}_{\mathbf{\φ}}$ 

Initialize

Loading Student:-VectorCalculus

Define the position vector R

$⟨\mathrm{\ρ} \cos \left(\mathrm{\theta }\right)\sin \left(\mathrm{\phi }\right)\,\mathrm{\ρ} \sin \left(\mathrm{\theta }\right)\sin \left(\mathrm{\phi }\right)\,\mathrm{\ρ} \cos \left(\mathrm{\phi }\right)⟩$$\stackrel{\text{assign to a name}}{\to }$$R$

Obtain a representation of ${\mathit{e}}_{\mathbf{\ρ}}$ 

$\mathbf{R}$ = $\stackrel{\text{differentiate}}{\to }$ $\stackrel{\text{assign to a name}}{\to }$$\mathrm{Er}$

Obtain a representation of ${\mathit{e}}_{\mathbf{\φ}}$ 

$\mathbf{R}$ = $\stackrel{\text{differentiate}}{\to }$ $\stackrel{\text{Euclidean-normalize}}{\to }$ $\stackrel{\text{assuming positive}}{\to }$ $\stackrel{\text{assign to a name}}{\to }$$\mathrm{Ef}$$$

Obtain a representation of ${\mathit{e}}_{\mathbf{\θ}}$ 

$\mathbf{R}$ = $\stackrel{\text{differentiate}}{\to }$ $\stackrel{\text{Euclidean-normalize}}{\to }$ $\stackrel{\text{simplify symbolic}}{\to }$ $\stackrel{\text{assign to a name}}{\to }$$\mathrm{Et}$

Solve for $\left\{\mathbf{i}\,\mathbf{j}\,\mathbf{k}\right\}$ in terms of  ${\mathit{e}}_{\mathbf{\ρ}}\,{\mathit{e}}_{\mathbf{\φ}}\,{\mathit{e}}_{\mathbf{\θ}}$ 

$\mathbf{Er}·⟨\mathbf{i}\,\mathbf{j}\,\mathbf{k}⟩\={\mathit{e}}_{\mathbf{\ρ}}\,\mathbf{Ef}·⟨\mathbf{i}\,\mathbf{j}\,\mathbf{k}⟩\={\mathit{e}}_{\mathbf{\φ}}\,\mathit{Et}·⟨\mathbf{i}\,\mathbf{j}\,\mathbf{k}⟩\={\mathit{e}}_{\mathbf{\θ}}$

$\cos \left(\mathrm{\θ}\right)\sin \left(\mathrm{\φ}\right)i\+\sin \left(\mathrm{\θ}\right)\sin \left(\mathrm{\φ}\right)j\+\cos \left(\mathrm{\φ}\right)k\=e_{\mathrm{\ρ}}\,\cos \left(\mathrm{\θ}\right)\cos \left(\mathrm{\φ}\right)i\+\sin \left(\mathrm{\θ}\right)\cos \left(\mathrm{\φ}\right)j-\sin \left(\mathrm{\φ}\right)k\=e_{\mathrm{\φ}}\,-\sin \left(\mathrm{\θ}\right)i\+\cos \left(\mathrm{\θ}\right)j\=e_{\mathrm{\θ}}$

$\stackrel{\text{solve (specified)}}{\to }$

$\left\{i\=-\frac{{\cos \left(\mathrm{\φ}\right)}^2\sin \left(\mathrm{\θ}\right)e_{\mathrm{\θ}}\+\sin \left(\mathrm{\θ}\right){\sin \left(\mathrm{\φ}\right)}^2{e}_{\mathrm{\θ}}-\cos \left(\mathrm{\φ}\right)\cos \left(\mathrm{\θ}\right)e_{\mathrm{\φ}}-e_{\mathrm{\ρ}}\cos \left(\mathrm{\θ}\right)\sin \left(\mathrm{\φ}\right)}{\left({\cos \left(\mathrm{\θ}\right)}^2\+{\sin \left(\mathrm{\θ}\right)}^2\right)\left({\cos \left(\mathrm{\φ}\right)}^2\+{\sin \left(\mathrm{\φ}\right)}^2\right)}\,j\=\frac{{\cos \left(\mathrm{\φ}\right)}^2\cos \left(\mathrm{\θ}\right)e_{\mathrm{\θ}}\+\cos \left(\mathrm{\θ}\right){\sin \left(\mathrm{\φ}\right)}^2{e}_{\mathrm{\θ}}\+\cos \left(\mathrm{\φ}\right)\sin \left(\mathrm{\θ}\right)e_{\mathrm{\φ}}\+\sin \left(\mathrm{\φ}\right)\sin \left(\mathrm{\θ}\right)e_{\mathrm{\ρ}}}{\left({\cos \left(\mathrm{\θ}\right)}^2\+{\sin \left(\mathrm{\θ}\right)}^2\right)\left({\cos \left(\mathrm{\φ}\right)}^2\+{\sin \left(\mathrm{\φ}\right)}^2\right)}\,k\=\frac{\cos \left(\mathrm{\φ}\right)e_{\mathrm{\ρ}}-\sin \left(\mathrm{\φ}\right)e_{\mathrm{\φ}}}{{\cos \left(\mathrm{\φ}\right)}^2\+{\sin \left(\mathrm{\φ}\right)}^2}\right\}$

$\stackrel{\text{simplify symbolic}}{\to }$

$\left\{i\=\cos \left(\mathrm{\φ}\right)\cos \left(\mathrm{\θ}\right)e_{\mathrm{\φ}}\+e_{\mathrm{\ρ}}\cos \left(\mathrm{\θ}\right)\sin \left(\mathrm{\φ}\right)-\sin \left(\mathrm{\θ}\right)e_{\mathrm{\θ}}\,j\=\cos \left(\mathrm{\φ}\right)\sin \left(\mathrm{\θ}\right)e_{\mathrm{\φ}}\+\sin \left(\mathrm{\φ}\right)\sin \left(\mathrm{\θ}\right)e_{\mathrm{\ρ}}\+\cos \left(\mathrm{\θ}\right)e_{\mathrm{\θ}}\,k\=\cos \left(\mathrm{\φ}\right)e_{\mathrm{\ρ}}-\sin \left(\mathrm{\φ}\right)e_{\mathrm{\φ}}\right\}$

$\stackrel{\text{assign to a name}}{\to }$

$S$

Change coordinates in the vector field F

$\genfrac{}{}{0ex}{}{a \mathbf{i}\+b \mathbf{j}\+c \mathbf{k}}{\phantom{x\=a}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{f(x)}}}{S}$

$a\left(\cos \left(\mathrm{\φ}\right)\cos \left(\mathrm{\θ}\right)e_{\mathrm{\φ}}\+e_{\mathrm{\ρ}}\cos \left(\mathrm{\θ}\right)\sin \left(\mathrm{\φ}\right)-\sin \left(\mathrm{\θ}\right)e_{\mathrm{\θ}}\right)\+b\left(\cos \left(\mathrm{\φ}\right)\sin \left(\mathrm{\θ}\right)e_{\mathrm{\φ}}\+\sin \left(\mathrm{\φ}\right)\sin \left(\mathrm{\θ}\right)e_{\mathrm{\ρ}}\+\cos \left(\mathrm{\θ}\right)e_{\mathrm{\θ}}\right)\+c\left(\cos \left(\mathrm{\φ}\right)e_{\mathrm{\ρ}}-\sin \left(\mathrm{\φ}\right)e_{\mathrm{\φ}}\right)$

$\stackrel{\text{collect w.r.t. e[rho]}}{\=}$

$\left(a\cos \left(\mathrm{\θ}\right)\sin \left(\mathrm{\φ}\right)\+b\sin \left(\mathrm{\θ}\right)\sin \left(\mathrm{\φ}\right)\+c\cos \left(\mathrm{\φ}\right)\right)e_{\mathrm{\ρ}}\+a\left(\cos \left(\mathrm{\φ}\right)\cos \left(\mathrm{\θ}\right)e_{\mathrm{\φ}}-\sin \left(\mathrm{\θ}\right)e_{\mathrm{\θ}}\right)\+b\left(\cos \left(\mathrm{\φ}\right)\sin \left(\mathrm{\θ}\right)e_{\mathrm{\φ}}\+\cos \left(\mathrm{\θ}\right)e_{\mathrm{\θ}}\right)-c\sin \left(\mathrm{\φ}\right)e_{\mathrm{\φ}}$

$\stackrel{\text{collect w.r.t. e[phi]}}{\=}$

$\left(a\cos \left(\mathrm{\θ}\right)\cos \left(\mathrm{\φ}\right)\+b\sin \left(\mathrm{\θ}\right)\cos \left(\mathrm{\φ}\right)-c\sin \left(\mathrm{\φ}\right)\right)e_{\mathrm{\φ}}\+\left(a\cos \left(\mathrm{\θ}\right)\sin \left(\mathrm{\φ}\right)\+b\sin \left(\mathrm{\θ}\right)\sin \left(\mathrm{\φ}\right)\+c\cos \left(\mathrm{\φ}\right)\right)e_{\mathrm{\ρ}}-a\sin \left(\mathrm{\θ}\right)e_{\mathrm{\θ}}\+b\cos \left(\mathrm{\θ}\right)e_{\mathrm{\θ}}$

$\stackrel{\text{collect w.r.t. e[theta]}}{\=}$

$\left(b\cos \left(\mathrm{\θ}\right)-a\sin \left(\mathrm{\θ}\right)\right)e_{\mathrm{\θ}}\+\left(a\cos \left(\mathrm{\θ}\right)\cos \left(\mathrm{\φ}\right)\+b\sin \left(\mathrm{\θ}\right)\cos \left(\mathrm{\φ}\right)-c\sin \left(\mathrm{\φ}\right)\right)e_{\mathrm{\φ}}\+\left(a\cos \left(\mathrm{\θ}\right)\sin \left(\mathrm{\φ}\right)\+b\sin \left(\mathrm{\θ}\right)\sin \left(\mathrm{\φ}\right)\+c\cos \left(\mathrm{\φ}\right)\right)e_{\mathrm{\ρ}}$

Initialize

$$\begin{gathered} \mathrm{with}\left(\mathrm{Student}:-\mathrm{VectorCalculus}\right)\: \\ \mathrm{BasisFormat}\left(\mathrm{false}\right)\: \end{gathered}$$

Define the vector field F

$\mathbf{F}≔\mathrm{VectorField}\left(⟨a\,b\,c⟩\right)\:$

Change to spherical coordinates by applying the MapToBasis command

$\mathrm{MapToBasis}\left(\mathbf{F}\,\mathrm{spherical}\left[\mathrm{\ρ}\,\mathrm{\φ}\,\mathrm{\θ}\right]\right)$

Define R, the position vector in spherical coordinates

$\mathbf{R}≔\mathrm{PositionVector}\left(\left[\mathrm{\rho }\,\mathrm{\phi }\,\mathrm{\theta }\right]\,\mathrm{spherical}\left[\mathrm{\rho }\,\mathrm{\phi }\,\mathrm{\theta }\right]\right)$

By differentiation and normalization, obtain unit tangent vectors along the spherical coordinate curves

$\mathbf{Er}≔\mathrm{diff}\left(\mathbf{R}\,\mathrm{\ρ}\right)$

$\mathbf{Ef}≔\mathrm{Normalize}\left(\mathrm{diff}\left(R\,\mathrm{\phi }\right)\right) assuming \mathrm{\ρ}\>0$

$\mathbf{Et}≔\mathrm{Normalize}\left(\mathrm{diff}\left(\mathbf{R}\,\mathrm{\θ}\right)\right) assuming \mathrm{\ρ}\>0\,\mathrm{\φ}\colon\colon \mathrm{RealRange}\left(0\,\mathrm{\π}\right)$

Express these unit vectors in terms of $\left\{\mathbf{i}\,\mathbf{j}\,\mathbf{k}\right\}$, equate to the names ${\mathit{e}}_{\mathbf{\ρ}}\,{\mathit{e}}_{\mathbf{\φ}}\,{\mathit{e}}_{\mathbf{\θ}}$ and solve for $\left\{\mathbf{i}\,\mathbf{j}\,\mathbf{k}\right\}$

$S≔\mathrm{simplify}\left(\mathrm{solve}\left(\left\{\mathbf{Er}·⟨\mathbf{i}\,\mathbf{j}\,\mathbf{k}⟩\={\mathit{e}}_{\mathbf{\ρ}}\,\mathbf{Ef}·⟨\mathbf{i}\,\mathbf{j}\,\mathbf{k}⟩\={\mathit{e}}_{\mathbf{\phi }}\,\mathbf{Et}·⟨\mathbf{i}\,\mathbf{j}\,\mathbf{k}⟩\={\mathit{e}}_{\mathbf{\theta }}\right\}\,\left\{\mathbf{i}\,\mathbf{j}\,\mathbf{k}\right\}\right)\right)$

$\left\{i\=\cos \left(\mathrm{\φ}\right)\cos \left(\mathrm{\θ}\right)e_{\mathrm{\φ}}\+e_{\mathrm{\ρ}}\cos \left(\mathrm{\θ}\right)\sin \left(\mathrm{\φ}\right)-\sin \left(\mathrm{\θ}\right)e_{\mathrm{\θ}}\,j\=\cos \left(\mathrm{\φ}\right)\sin \left(\mathrm{\θ}\right)e_{\mathrm{\φ}}\+\sin \left(\mathrm{\θ}\right)\sin \left(\mathrm{\φ}\right)e_{\mathrm{\ρ}}\+\cos \left(\mathrm{\θ}\right)e_{\mathrm{\θ}}\,k\=\cos \left(\mathrm{\φ}\right)e_{\mathrm{\ρ}}-\sin \left(\mathrm{\φ}\right)e_{\mathrm{\φ}}\right\}$

In the field $\mathbf{F}\=a \mathbf{i}\+b \mathbf{j}\+c \mathbf{k}$, replace $\left\{\mathbf{i}\,\mathbf{j}\,\mathbf{k}\right\}$ with their equivalents in terms of  ${\mathit{e}}_{\mathbf{\ρ}}\,{\mathit{e}}_{\mathbf{\φ}}\,{\mathit{e}}_{\mathbf{\θ}}$

$\mathrm{collect}\left(\mathrm{eval}\left(a \mathbf{i}\+b \mathbf{j}\+c \mathbf{k}\,S\right)\,\left[{\mathit{e}}_{\mathbf{\ρ}}\,{\mathit{e}}_{\mathbf{\phi }}\,{\mathit{e}}_{\mathbf{\theta }}\right]\right)$

$\left(a\sin \left(\mathrm{\φ}\right)\cos \left(\mathrm{\θ}\right)\+b\sin \left(\mathrm{\θ}\right)\sin \left(\mathrm{\φ}\right)\+c\cos \left(\mathrm{\φ}\right)\right)e_{\mathrm{\ρ}}\+\left(a\cos \left(\mathrm{\θ}\right)\cos \left(\mathrm{\φ}\right)\+b\sin \left(\mathrm{\θ}\right)\cos \left(\mathrm{\φ}\right)-c\sin \left(\mathrm{\φ}\right)\right)e_{\mathrm{\φ}}\+\left(-a\sin \left(\mathrm{\θ}\right)\+b\cos \left(\mathrm{\θ}\right)\right)e_{\mathrm{\θ}}$

Compare

$\mathrm{MapToBasis}\left(\mathbf{F}\,\mathrm{spherical}\left[\mathrm{\ρ}\,\mathrm{\φ}\,\mathrm{\θ}\right]\right)$ = $$