The Jacobian(f, v) command computes the Jacobian Matrix of a list or Vector of expressions f with respect to the variables in v.

If the point p is supplied, the computed Jacobian will be evaluated at that point. The dimension of the point must equal the number of variables in v.

If the right side of det is true, an expression sequence containing the Jacobian Matrix and its determinant, in that order, is returned. If the right side of det is false, the Jacobian Matrix is returned. If this parameter is the word determinant, it is interpreted as determinant=true. If the det parameter is not specified, it defaults to determinant=false.

If the computation of the determinant is requested, then the number of component functions given in f must be the same as the number of variables given in v.

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

$\mathrm{Jacobian}\left(\left[r\cos \left(t\right)\,r\sin \left(t\right)\,r^{2}t\right]\,\left[r\,t\right]\right)$

$\left[\begin{array}{cc}\cos \left(t\right)& -r\sin \left(t\right)\\ \sin \left(t\right)& r\cos \left(t\right)\\ 2rt& r^2\end{array}\right]$

$M,d≔\mathrm{Jacobian}\left(\left[r\sin \left(\mathrm{\phi }\right)\cos \left(\mathrm{\theta }\right)\,r\sin \left(\mathrm{\phi }\right)\sin \left(\mathrm{\theta }\right)\,r\cos \left(\mathrm{\phi }\right)\right]\,\left[r\,\mathrm{\phi }\,\mathrm{\theta }\right]\,'\mathrm{determinant}'=\mathrm{true}\right)$

$M,d≔\left[\begin{array}{ccc}\sin \left(\mathrm{\phi }\right)\cos \left(\mathrm{\theta }\right)& r\cos \left(\mathrm{\phi }\right)\cos \left(\mathrm{\theta }\right)& -r\sin \left(\mathrm{\phi }\right)\sin \left(\mathrm{\theta }\right)\\ \sin \left(\mathrm{\phi }\right)\sin \left(\mathrm{\theta }\right)& r\cos \left(\mathrm{\phi }\right)\sin \left(\mathrm{\theta }\right)& r\sin \left(\mathrm{\phi }\right)\cos \left(\mathrm{\theta }\right)\\ \cos \left(\mathrm{\phi }\right)& -r\sin \left(\mathrm{\phi }\right)& 0\end{array}\right],{\sin \left(\mathrm{\phi }\right)}^3{\cos \left(\mathrm{\theta }\right)}^2{r}^2+{\sin \left(\mathrm{\phi }\right)}^3{\sin \left(\mathrm{\theta }\right)}^2{r}^2+\sin \left(\mathrm{\phi }\right){\cos \left(\mathrm{\theta }\right)}^2{\cos \left(\mathrm{\phi }\right)}^2{r}^2+\sin \left(\mathrm{\phi }\right){\sin \left(\mathrm{\theta }\right)}^2{\cos \left(\mathrm{\phi }\right)}^2{r}^2$

$\mathrm{simplify}\left(d\,\mathrm{trig}\right)$

$r^2\sin \left(\mathrm{\phi }\right)$

$\mathrm{Jacobian}\left(⟨x^2+y\,2y⟩\,\left[x\,y\right]=\left[-1\,1\right]\right)$

$\left[\begin{array}{cc}\mathrm{-2}& 1\\ 0& 2\end{array}\right]$