Question

Ayuda por favor.
Sean f: R² -> R² y g: R³ -> R², dos campos vectoriales definidos por: f(x,y) = [$e^{x+2y}$, sen(y + 2x)], g(u,v,w) = (u + 2v² + 3w³, 2v - u²)

a) Halle las diferenciales Df(x, y) y Dg(u, v, w)
b) Halle la diferencial Dh(1, -1, 1) para la función h(u, v, w) = f [g(u, v, w)]

Answer 1

(a.1)

$Df(x,y)=\left[\begin{matrix} \nabla(e^{x+2y})\\\nabla\sin(y+2x) \end{matrix}\right]\\ \\ \\ Df(x,y)=\left[\begin{matrix} e^{x+2y}&2e^{x+2y}\\2\cos(y+2x)&\cos(y+2x) \end{matrix}\right]$

(a.2)
$Dg(u,v,w)=\left[\begin{matrix} \nabla(u+2v^2+3w^3)\\ \nabla(2v-u^2) \end{matrix}\right]\\ \\ \\ Dg(u,v,w)=\left[\begin{matrix} 1&4v&9w^2\\ -2u&2&0 \end{matrix}\right]$

(b)

$Dh(1,-1,1)=J_h(1,-1,1)=J_f(g(1,-1,1))\cdot J_g(1,-1,1)\\ \\ \\ Dh(1,-1,1)=J_f(6,-3)\cdot J_g(1,-1,1)\\ \\ \\ Dh(1,-1,1)=\left[\begin{matrix} 1&2\\ 2&1 \end{matrix}\right] \left[\begin{matrix} 1&-4&9\\ -2&2&0 \end{matrix}\right]\\ \\ \\ Dh(1,-1,1)=\left[\begin{matrix} -3&0&9\\ 0&-6&18 \end{matrix}\right]$