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VectorCalculus

 Jacobian
 computes the Jacobian Matrix of a list or Vector of expressions

 Calling Sequence Jacobian(f, v, det) Jacobian(f, v=p, det)

Parameters

 f - list(algebraic) or Vector(algebraic); expressions to be differentiated v - (optional) list(name); specify the variables of differentiation p - (optional) list(algebraic); point at which the Jacobian is evaluated det - (optional) equation of the form determinant=t, where t is either true or false; specify whether to return the determinant

Description

 • 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 v is not provided, the differentiation variables are determined from the coordinate system of f, if f is a Vector, and otherwise from the ambient coordinate system (see SetCoordinates).
 • If p is supplied, the computed Jacobian will be evaluated at the corresponding point. The dimension of the point must equal the number of differentiation variables.
 • If the right side of det is true, an expression sequence containing the Jacobian Matrix and its determinant 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.

Examples

 > $\mathrm{with}\left(\mathrm{VectorCalculus}\right):$
 > $\mathrm{Jacobian}\left(\left[r\mathrm{cos}\left(t\right),r\mathrm{sin}\left(t\right),{r}^{2}t\right],\left[r,t\right]\right)$
 $\left[\begin{array}{cc}{\mathrm{cos}}{}\left({t}\right)& {-}{r}{}{\mathrm{sin}}{}\left({t}\right)\\ {\mathrm{sin}}{}\left({t}\right)& {r}{}{\mathrm{cos}}{}\left({t}\right)\\ {2}{}{r}{}{t}& {{r}}^{{2}}\end{array}\right]$ (1)
 > $M,d≔\mathrm{Jacobian}\left(\left[r\mathrm{sin}\left(\mathrm{φ}\right)\mathrm{cos}\left(\mathrm{θ}\right),r\mathrm{sin}\left(\mathrm{φ}\right)\mathrm{sin}\left(\mathrm{θ}\right),r\mathrm{cos}\left(\mathrm{φ}\right)\right],\left[r,\mathrm{φ},\mathrm{θ}\right],'\mathrm{determinant}'\right)$
 ${M}{,}{d}{≔}\left[\begin{array}{ccc}{\mathrm{sin}}{}\left({\mathrm{φ}}\right){}{\mathrm{cos}}{}\left({\mathrm{θ}}\right)& {r}{}{\mathrm{cos}}{}\left({\mathrm{φ}}\right){}{\mathrm{cos}}{}\left({\mathrm{θ}}\right)& {-}{r}{}{\mathrm{sin}}{}\left({\mathrm{φ}}\right){}{\mathrm{sin}}{}\left({\mathrm{θ}}\right)\\ {\mathrm{sin}}{}\left({\mathrm{φ}}\right){}{\mathrm{sin}}{}\left({\mathrm{θ}}\right)& {r}{}{\mathrm{cos}}{}\left({\mathrm{φ}}\right){}{\mathrm{sin}}{}\left({\mathrm{θ}}\right)& {r}{}{\mathrm{sin}}{}\left({\mathrm{φ}}\right){}{\mathrm{cos}}{}\left({\mathrm{θ}}\right)\\ {\mathrm{cos}}{}\left({\mathrm{φ}}\right)& {-}{r}{}{\mathrm{sin}}{}\left({\mathrm{φ}}\right)& {0}\end{array}\right]{,}{{\mathrm{sin}}{}\left({\mathrm{φ}}\right)}^{{3}}{}{{\mathrm{cos}}{}\left({\mathrm{θ}}\right)}^{{2}}{}{{r}}^{{2}}{+}{{\mathrm{sin}}{}\left({\mathrm{φ}}\right)}^{{3}}{}{{\mathrm{sin}}{}\left({\mathrm{θ}}\right)}^{{2}}{}{{r}}^{{2}}{+}{\mathrm{sin}}{}\left({\mathrm{φ}}\right){}{{\mathrm{cos}}{}\left({\mathrm{θ}}\right)}^{{2}}{}{{\mathrm{cos}}{}\left({\mathrm{φ}}\right)}^{{2}}{}{{r}}^{{2}}{+}{\mathrm{sin}}{}\left({\mathrm{φ}}\right){}{{\mathrm{sin}}{}\left({\mathrm{θ}}\right)}^{{2}}{}{{\mathrm{cos}}{}\left({\mathrm{φ}}\right)}^{{2}}{}{{r}}^{{2}}$ (2)
 > $\mathrm{simplify}\left(d,\mathrm{trig}\right)$
 ${\mathrm{sin}}{}\left({\mathrm{φ}}\right){}{{r}}^{{2}}$ (3)
 > $\mathrm{Jacobian}\left(⟨{x}^{2}+y,2y⟩,\left[x,y\right]=\left[-1,1\right]\right)$
 $\left[\begin{array}{rr}{-}{2}& {1}\\ {0}& {2}\end{array}\right]$ (4)
 > $\mathrm{Jacobian}\left(\mathrm{Vector}\left(\left[{r}^{2},rt\right],'\mathrm{coordinates}'={'\mathrm{polar}'}_{r,t}\right)\right)$
 $\left[\begin{array}{cc}{2}{}{r}& {0}\\ {t}& {r}\end{array}\right]$ (5)
 > $\mathrm{SetCoordinates}\left({'\mathrm{cylindrical}'}_{r,\mathrm{θ},z}\right)$
 ${{\mathrm{cylindrical}}}_{{r}{,}{\mathrm{θ}}{,}{z}}$ (6)
 > $M,d≔\mathrm{Jacobian}\left(\left[r\mathrm{θ},{\mathrm{θ}}^{2}z,z\right],'\mathrm{determinant}'\right)$
 ${M}{,}{d}{≔}\left[\begin{array}{ccc}{\mathrm{θ}}& {r}& {0}\\ {0}& {2}{}{\mathrm{θ}}{}{z}& {{\mathrm{θ}}}^{{2}}\\ {0}& {0}& {1}\end{array}\right]{,}{2}{}{{\mathrm{θ}}}^{{2}}{}{z}$ (7)
 > $M,d≔\mathrm{Jacobian}\left(\left[art,tw,{w}^{2}a\right],'\mathrm{determinant}',\left[w,t,a\right]=\left[-1,1,2\right]\right)$
 ${M}{,}{d}{≔}\left[\begin{array}{ccc}{0}& {2}{}{r}& {r}\\ {1}& {-}{1}& {0}\\ {-}{4}& {0}& {1}\end{array}\right]{,}{-}{6}{}{r}$ (8)