91Ó°ÊÓ

Compute the partial derivatives of x, y, and z with respect to u, v, and w. $$\frac{\partial x}{\partial u} = 0, \frac{\partial x}{\partial v} = w, \frac{\partial x}{\partial w} = v$$ $$\frac{\partial y}{\partial u} = w, \frac{\partial y}{\partial v} = 0, \frac{\partial y}{\partial w} = u$$ $$\frac{\partial z}{\partial u} = 2u, \frac{\partial z}{\partial v} = -2v, \frac{\partial z}{\partial w} = 0$$

Using the partial derivatives we computed in Step 1, write out the Jacobian matrix: $$J(u, v, w) = \begin{bmatrix} 0 & w & v \\ w & 0 & u \\ 2u & -2v & 0 \end{bmatrix}$$

In order to evaluate the Jacobian, compute the determinant of the matrix: $$|J(u, v, w)| = \bigg| \begin{bmatrix} 0 & w & v \\ w & 0 & u \\ 2u & -2v & 0 \end{bmatrix} \bigg| = 0 \times \bigg| \begin{bmatrix} 0 & u \\ -2v & 0 \end{bmatrix} \bigg| - w \times \bigg| \begin{bmatrix} w & u \\ 2u & 0 \\ \end{bmatrix} \bigg| + v \times \bigg| \begin{bmatrix} w & 0 \\ 2u & -2v \\ \end{bmatrix} \bigg|$$ Now, compute the three subdeterminants: $$\bigg| \begin{bmatrix} 0 & u \\ -2v & 0 \end{bmatrix} \bigg| = 0$$ $$\bigg| \begin{bmatrix} w & u \\ 2u & 0 \\ \end{bmatrix} \bigg| = -2wu^2$$ $$\bigg| \begin{bmatrix} w & 0 \\ 2u & -2v \\ \end{bmatrix} \bigg| = -2wv^2$$ Now, substitute the subdeterminants back into the expression for the determinant: $$|J(u, v, w)| = 0 - w(-2wu^2) + v(-2wv^2)$$ Simplify the expression: $$|J(u, v, w)| = 2w^2u^2 + 2v^2w^2$$ Therefore, the Jacobian J(u, v, w) is: $$|J(u, v, w)| = 2w^2(u^2 + v^2)$$