Jacobian of three functions
If $$F=xu+v-y$$ $$G=u^2+\cos y+ w$$ $$H=zu-v+vw$$ the question is to find the Jacobian of $(F,G,H)$ wrt $(u,v,w)$
I am aware of the determinant expression for the Jacobian.But couldn't apply here since here the functions involve product of variables as well.Any idea on how to solve these type of problem shall be highly appreciated.Thanks.
$\endgroup$ 11 Answer
$\begingroup$Consider a differentiable mapping
$$ f \, : \, U \subset \mathbb{R}^3 \, \rightarrow \, \mathbb{R}^3 $$
such that, for all $(u,v,w) \in U$ :
$$ f(u,v,w) = \big( F(u,v,w), G(u,v,w), H(u,v,w) \big). $$
$F$, $G$ or $H$ are real-valued coordinate functions. The Jacobian matrix of $f$ at $x_0 = (u_0, v_0, w_0) \in U$ is defined as :
$$ \mathrm{Jac}(f,x_0) = \begin{bmatrix} \displaystyle \frac{\partial F}{\partial u}(x_0) & \displaystyle \frac{\partial F}{\partial v}(x_0) & \displaystyle \frac{\partial F}{\partial w}(x_0) \\[2mm] \displaystyle \frac{\partial G}{\partial u}(x_0) & \displaystyle \frac{\partial G}{\partial v}(x_0) & \displaystyle \frac{\partial G}{\partial w}(x_0) \\[2mm] \displaystyle \frac{\partial H}{\partial u}(x_0) & \displaystyle \frac{\partial H}{\partial v}(x_0) & \displaystyle \frac{\partial H}{\partial w}(x_0) \end{bmatrix} $$
In your case, with $F(u,v,w) = xu + v - y$, $G(u,v,w) = u^2 + \cos(y) + w$ and $H(u,v,w) = zu - v + vw$, you obtain:
$$ \mathrm{Jac}(f,x_0) = \begin{bmatrix} x & 1 & 0 \\ 2u_0 & 0 & 1 \\ z & w_0 - 1 & v_0 \end{bmatrix}. $$
$\endgroup$
