$\left[\sin \left(\mathrm{\phi }\right)\sin \left(\mathrm{\theta }\right)\+\sin \left(\mathrm{\phi }\right)\cos \left(\mathrm{\theta }\right)-\cos \left(\mathrm{\phi }\right)\right] \cos \left(\mathrm{\φ}\right)$ 
$-\left[\cos \left(\mathrm{\phi }\right)\sin \left(\mathrm{\theta }\right)\+\cos \left(\mathrm{\phi }\right)\cos \left(\mathrm{\theta }\right)\+\sin \left(\mathrm{\phi }\right)\right] \sin \left(\mathrm{\phi }\right)$

$\= 0\+0-\left({\cos }^2\left(\mathrm{\φ}\right)\+{\sin }^2\left(\mathrm{\φ}\right)\right)\= -1$$$

Initialize

Loading Student:-VectorCalculus

Define the Cartesian vector field F

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

Obtain the curl of F

$\nabla \times \mathbf{F}$ = $$

Alternate calculation of the curl of F

$\mathbf{F}$ = $\stackrel{\text{curl}}{\to }$ $$

Change F to spherical coordinates

$\mathbf{F}$ = $\left[\begin{array}{c}y\\ -z\\ -x\end{array}\right]$$\stackrel{\text{change coordinates}}{\to }$$\left[\begin{array}{c}\mathrm{\rho }{\sin \left(\mathrm{\phi }\right)}^2\sin \left(\mathrm{\theta }\right)\cos \left(\mathrm{\theta }\right)-\mathrm{\rho }\sin \left(\mathrm{\phi }\right)\cos \left(\mathrm{\theta }\right)\cos \left(\mathrm{\phi }\right)-\mathrm{\rho }\cos \left(\mathrm{\phi }\right)\sin \left(\mathrm{\theta }\right)\sin \left(\mathrm{\phi }\right)\\ \mathrm{\rho }\sin \left(\mathrm{\phi }\right)\sin \left(\mathrm{\theta }\right)\cos \left(\mathrm{\theta }\right)\cos \left(\mathrm{\phi }\right)+\mathrm{\rho }{\sin \left(\mathrm{\phi }\right)}^2\cos \left(\mathrm{\theta }\right)-\mathrm{\rho }{\cos \left(\mathrm{\phi }\right)}^2\sin \left(\mathrm{\theta }\right)\\ -\mathrm{\rho }\sin \left(\mathrm{\phi }\right){\sin \left(\mathrm{\theta }\right)}^2-\mathrm{\rho }\cos \left(\mathrm{\phi }\right)\cos \left(\mathrm{\theta }\right)\end{array}\right]$$\stackrel{\text{assign to a name}}{\to }$$G$$$

Obtain the curl in spherical coordinates

$\nabla \times \mathbf{G}$ = $\left[\begin{array}{c}\frac{\mathrm{\rho }\cos \left(\mathrm{\phi }\right)\left(-\mathrm{\rho }\sin \left(\mathrm{\phi }\right){\sin \left(\mathrm{\theta }\right)}^2-\mathrm{\rho }\cos \left(\mathrm{\phi }\right)\cos \left(\mathrm{\theta }\right)\right)+\mathrm{\rho }\sin \left(\mathrm{\phi }\right)\left(-\mathrm{\rho }\cos \left(\mathrm{\phi }\right){\sin \left(\mathrm{\theta }\right)}^2+\mathrm{\rho }\sin \left(\mathrm{\phi }\right)\cos \left(\mathrm{\theta }\right)\right)-\mathrm{\rho }\left(\mathrm{\rho }\sin \left(\mathrm{\phi }\right){\cos \left(\mathrm{\theta }\right)}^2\cos \left(\mathrm{\phi }\right)-\mathrm{\rho }\sin \left(\mathrm{\phi }\right){\sin \left(\mathrm{\theta }\right)}^2\cos \left(\mathrm{\phi }\right)-\mathrm{\rho }{\sin \left(\mathrm{\phi }\right)}^2\sin \left(\mathrm{\theta }\right)-\mathrm{\rho }{\cos \left(\mathrm{\phi }\right)}^2\cos \left(\mathrm{\theta }\right)\right)}{{\mathrm{\rho }}^2\sin \left(\mathrm{\phi }\right)}\\ \frac{\mathrm{\rho }{\sin \left(\mathrm{\phi }\right)}^2{\cos \left(\mathrm{\theta }\right)}^2-\mathrm{\rho }{\sin \left(\mathrm{\phi }\right)}^2{\sin \left(\mathrm{\theta }\right)}^2+\mathrm{\rho }\cos \left(\mathrm{\phi }\right)\sin \left(\mathrm{\theta }\right)\sin \left(\mathrm{\phi }\right)-\mathrm{\rho }\sin \left(\mathrm{\phi }\right)\cos \left(\mathrm{\theta }\right)\cos \left(\mathrm{\phi }\right)-\sin \left(\mathrm{\phi }\right)\left(-\mathrm{\rho }\sin \left(\mathrm{\phi }\right){\sin \left(\mathrm{\theta }\right)}^2-\mathrm{\rho }\cos \left(\mathrm{\phi }\right)\cos \left(\mathrm{\theta }\right)\right)-\mathrm{\rho }\sin \left(\mathrm{\phi }\right)\left(-\sin \left(\mathrm{\phi }\right){\sin \left(\mathrm{\theta }\right)}^2-\cos \left(\mathrm{\phi }\right)\cos \left(\mathrm{\theta }\right)\right)}{\mathrm{\rho }\sin \left(\mathrm{\phi }\right)}\\ \frac{-\mathrm{\rho }\sin \left(\mathrm{\phi }\right)\sin \left(\mathrm{\theta }\right)\cos \left(\mathrm{\theta }\right)\cos \left(\mathrm{\phi }\right)+\mathrm{\rho }\left(\sin \left(\mathrm{\phi }\right)\sin \left(\mathrm{\theta }\right)\cos \left(\mathrm{\theta }\right)\cos \left(\mathrm{\phi }\right)+{\sin \left(\mathrm{\phi }\right)}^2\cos \left(\mathrm{\theta }\right)-{\cos \left(\mathrm{\phi }\right)}^2\sin \left(\mathrm{\theta }\right)\right)+\mathrm{\rho }{\cos \left(\mathrm{\phi }\right)}^2\cos \left(\mathrm{\theta }\right)-\mathrm{\rho }{\sin \left(\mathrm{\phi }\right)}^2\sin \left(\mathrm{\theta }\right)}{\mathrm{\rho }}\end{array}\right]$$\stackrel{\text{simplify}}{\=}$$\left[\begin{array}{c}\left(\cos \left(\mathrm{\theta }\right)+\sin \left(\mathrm{\theta }\right)\right)\sin \left(\mathrm{\phi }\right)-\cos \left(\mathrm{\phi }\right)\\ \left(\cos \left(\mathrm{\theta }\right)+\sin \left(\mathrm{\theta }\right)\right)\cos \left(\mathrm{\phi }\right)+\sin \left(\mathrm{\phi }\right)\\ -\sin \left(\mathrm{\theta }\right)+\cos \left(\mathrm{\theta }\right)\end{array}\right]$$\stackrel{\text{assign to a name}}{\to }$$\mathrm{curlG}$$$

Change the coordinates in $\nabla \times \mathbf{G}$ back to Cartesian

$\mathrm{curlG}$ = $\left[\begin{array}{c}\left(\cos \left(\mathrm{\theta }\right)+\sin \left(\mathrm{\theta }\right)\right)\sin \left(\mathrm{\phi }\right)-\cos \left(\mathrm{\phi }\right)\\ \left(\cos \left(\mathrm{\theta }\right)+\sin \left(\mathrm{\theta }\right)\right)\cos \left(\mathrm{\phi }\right)+\sin \left(\mathrm{\phi }\right)\\ -\sin \left(\mathrm{\theta }\right)+\cos \left(\mathrm{\theta }\right)\end{array}\right]$$\stackrel{\text{change coordinates}}{\to }$$\left[\begin{array}{c}\frac{\left(\frac{\left(\frac{x}{\sqrt{x^2+y^2}}+\frac{y}{\sqrt{x^2+y^2}}\right)\sqrt{x^2+y^2}}{\sqrt{x^2+y^2+z^2}}-\frac{z}{\sqrt{x^2+y^2+z^2}}\right)x}{\sqrt{x^2+y^2+z^2}}+\frac{\left(\frac{\left(\frac{x}{\sqrt{x^2+y^2}}+\frac{y}{\sqrt{x^2+y^2}}\right)z}{\sqrt{x^2+y^2+z^2}}+\frac{\sqrt{x^2+y^2}}{\sqrt{x^2+y^2+z^2}}\right)xz}{\sqrt{x^2+y^2}\sqrt{x^2+y^2+z^2}}-\frac{\left(-\frac{y}{\sqrt{x^2+y^2}}+\frac{x}{\sqrt{x^2+y^2}}\right)y}{\sqrt{x^2+y^2}}\\ \frac{\left(\frac{\left(\frac{x}{\sqrt{x^2+y^2}}+\frac{y}{\sqrt{x^2+y^2}}\right)\sqrt{x^2+y^2}}{\sqrt{x^2+y^2+z^2}}-\frac{z}{\sqrt{x^2+y^2+z^2}}\right)y}{\sqrt{x^2+y^2+z^2}}+\frac{\left(\frac{\left(\frac{x}{\sqrt{x^2+y^2}}+\frac{y}{\sqrt{x^2+y^2}}\right)z}{\sqrt{x^2+y^2+z^2}}+\frac{\sqrt{x^2+y^2}}{\sqrt{x^2+y^2+z^2}}\right)yz}{\sqrt{x^2+y^2}\sqrt{x^2+y^2+z^2}}+\frac{\left(-\frac{y}{\sqrt{x^2+y^2}}+\frac{x}{\sqrt{x^2+y^2}}\right)x}{\sqrt{x^2+y^2}}\\ \frac{\left(\frac{\left(\frac{x}{\sqrt{x^2+y^2}}+\frac{y}{\sqrt{x^2+y^2}}\right)\sqrt{x^2+y^2}}{\sqrt{x^2+y^2+z^2}}-\frac{z}{\sqrt{x^2+y^2+z^2}}\right)z}{\sqrt{x^2+y^2+z^2}}+\frac{\left(\frac{\left(\frac{x}{\sqrt{x^2+y^2}}+\frac{y}{\sqrt{x^2+y^2}}\right)z}{\sqrt{x^2+y^2+z^2}}+\frac{\sqrt{x^2+y^2}}{\sqrt{x^2+y^2+z^2}}\right)\left(-x^2-y^2\right)}{\sqrt{x^2+y^2}\sqrt{x^2+y^2+z^2}}\end{array}\right]$$\stackrel{\text{simplify}}{\=}$ $$

Initialize

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

Define the vector field F

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

Obtain the curl of F

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

Apply the MapToBasis command to change F to spherical coordinates

$\mathbf{G}≔\mathrm{MapToBasis}\left(\mathbf{F}\,\mathrm{spherical}\left[\mathrm{\rho }\,\mathrm{\phi }\,\mathrm{\theta }\right]\right)$

Obtain the curl of G, that is the curl of F in spherical coordinates

$\mathbf{curlG}≔\mathrm{simplify}\left(\mathrm{Curl}\left(\mathbf{G}\right)\right)$

Restore Cartesian coordinates in $\nabla \times \mathbf{G}$ 

$\mathrm{simplify}\left(\mathrm{MapToBasis}\left(\mathbf{curlG}\,\mathrm{cartesian}\left[x\,y\,z\right]\right)\right)$