Ayuda por favor.
Sean f: R² -> R² y g: R³ -> R², dos campos vectoriales definidos por: f(x,y) = [
, 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)]
Respuestas a la pregunta
Contestado por
2
(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]\\ \\ \\](https://tex.z-dn.net/?f=Df%28x%2Cy%29%3D%5Cleft%5B%5Cbegin%7Bmatrix%7D%0A%5Cnabla%28e%5E%7Bx%2B2y%7D%29%5C%5C%5Cnabla%5Csin%28y%2B2x%29%0A%5Cend%7Bmatrix%7D%5Cright%5D%5C%5C+%5C%5C+%5C%5C%0ADf%28x%2Cy%29%3D%5Cleft%5B%5Cbegin%7Bmatrix%7D%0Ae%5E%7Bx%2B2y%7D%26amp%3B2e%5E%7Bx%2B2y%7D%5C%5C2%5Ccos%28y%2B2x%29%26amp%3B%5Ccos%28y%2B2x%29%0A%5Cend%7Bmatrix%7D%5Cright%5D%5C%5C+%5C%5C+%5C%5C "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]\\ \\ \\](https://tex.z-dn.net/?f=Dg%28u%2Cv%2Cw%29%3D%5Cleft%5B%5Cbegin%7Bmatrix%7D%0A%5Cnabla%28u%2B2v%5E2%2B3w%5E3%29%5C%5C%0A%5Cnabla%282v-u%5E2%29%0A%5Cend%7Bmatrix%7D%5Cright%5D%5C%5C+%5C%5C+%5C%5C%0ADg%28u%2Cv%2Cw%29%3D%5Cleft%5B%5Cbegin%7Bmatrix%7D%0A1%26amp%3B4v%26amp%3B9w%5E2%5C%5C%0A-2u%26amp%3B2%26amp%3B0%0A%5Cend%7Bmatrix%7D%5Cright%5D%5C%5C+%5C%5C+%5C%5C "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]](https://tex.z-dn.net/?f=Dh%281%2C-1%2C1%29%3DJ_h%281%2C-1%2C1%29%3DJ_f%28g%281%2C-1%2C1%29%29%5Ccdot+J_g%281%2C-1%2C1%29%5C%5C+%5C%5C+%5C%5C%0ADh%281%2C-1%2C1%29%3DJ_f%286%2C-3%29%5Ccdot+J_g%281%2C-1%2C1%29%5C%5C+%5C%5C+%5C%5C%0ADh%281%2C-1%2C1%29%3D%5Cleft%5B%5Cbegin%7Bmatrix%7D%0A1%26amp%3B2%5C%5C%0A2%26amp%3B1%0A%5Cend%7Bmatrix%7D%5Cright%5D%0A%5Cleft%5B%5Cbegin%7Bmatrix%7D%0A1%26amp%3B-4%26amp%3B9%5C%5C%0A-2%26amp%3B2%26amp%3B0%0A%5Cend%7Bmatrix%7D%5Cright%5D%5C%5C+%5C%5C+%5C%5C%0ADh%281%2C-1%2C1%29%3D%5Cleft%5B%5Cbegin%7Bmatrix%7D%0A-3%26amp%3B0%26amp%3B9%5C%5C%0A0%26amp%3B-6%26amp%3B18%0A%5Cend%7Bmatrix%7D%5Cright%5D%0A "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]")
(a.2)
(b)
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