91Ó°ÊÓ

These are the given transformations: $$x = v + w$$ $$y = u + w$$ $$z = u + v$$

We need to compute the following partial derivatives: $$\frac{\partial x}{\partial u}, \ \frac{\partial x}{\partial v}, \ \frac{\partial x}{\partial w},$$ $$\frac{\partial y}{\partial u}, \ \frac{\partial y}{\partial v}, \ \frac{\partial y}{\partial w},$$ $$\frac{\partial z}{\partial u}, \ \frac{\partial z}{\partial v}, \ \frac{\partial z}{\partial w}.$$ Using the transformation equations, we can compute the partial derivatives as follows: $$\frac{\partial x}{\partial u} = 0, \ \frac{\partial x}{\partial v} = 1, \ \frac{\partial x}{\partial w} = 1,$$ $$\frac{\partial y}{\partial u} = 1, \ \frac{\partial y}{\partial v} = 0, \ \frac{\partial y}{\partial w} = 1,$$ $$\frac{\partial z}{\partial u} = 1, \ \frac{\partial z}{\partial v} = 1, \ \frac{\partial z}{\partial w} = 0.$$

The Jacobian matrix \(J(u, v, w)\) is: $$J(u, v, w) = \begin{bmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} & \frac{\partial x}{\partial w} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} & \frac{\partial y}{\partial w} \\ \frac{\partial z}{\partial u} & \frac{\partial z}{\partial v} & \frac{\partial z}{\partial w} \end{bmatrix} = \begin{bmatrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{bmatrix}$$ The Jacobian for this transformation is: $$J(u, v, w) = \begin{bmatrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{bmatrix}$$