\[\mathrm d\sin(xyz)= yz\cos(xyz)\mathrm dx +zx\cos(xyz)\mathrm dy +xy\cos(xyz)\mathrm dz\] d’où la matrice hessienne \[\left[ \begin{smallmatrix} -y^2z^2\sin(xyz) &z\cos(xyz) -xyz^2\sin(xyz)&y\cos(xyz)-xy^2z\sin(xyz) \\ z\cos(xyz)-xyz^2\sin(xyz)&-x^2z^2\sin(xyz) &x\cos(xyz)-x^2yz\sin(xyz) \\ y\cos(xyz)-xy^2z\sin(xyz)&y\cos(xyz) -x^2yz\sin(xyz)&-x^2y^2\sin(xyz) \end{smallmatrix} \right] .\] De même \[\begin{align*} \mathrm d(\sin^2(y/x))&= -{2y}{x^{-2}}\sin(y/x)\cos(y/x)\mathrm dx + 2x^{-1} \sin(y/x)\cos(y/x) \mathrm dy \\ &= \sin(2y/x)\left( -\frac y {x^2} dx + \frac 1x dy\right) \end{align*}\] d’où la matrice hessienne \[\left[ \begin{matrix} {2y}{x^{-3}}\sin(2y/x)+ {2y^2}{x^{-4}}\cos(2y/x) & -{x^{-2}}\sin(2y/x) -{2y}{x^{-3}}\cos(2y/x) \\ -x^{-2} \sin(2y/x) -2yx^{-3} \cos(2y/x)& 2x^{-2} \cos(2y/x) \end{matrix} \right] .\]

exo7 2638