Problem 39 - Calculus Early Transcendentals

Question

Evaluate the Jacobians \(J(u, v, w)\) for the following transformations. $$x=v w, y=u w, z=u^{2}-v^{2}$$

Step-by-Step Solution

Answer

Question: Determine the Jacobian, J(u, v, w), given the transformations below: x(u, v, w) = vw y(u, v, w) = uw z(u, v, w) = u^2 - v^2 Answer: J(u, v, w) = 4uvw

Step 1: Calculate Partial Derivatives

Compute the partial derivatives of the given transformations: \(\frac{\partial x}{\partial u}, \frac{\partial x}{\partial v}, \frac{\partial x}{\partial w}, \frac{\partial y}{\partial u}, \frac{\partial y}{\partial v}, \frac{\partial y}{\partial w}, \frac{\partial z}{\partial u}, \frac{\partial z}{\partial v}, \frac{\partial z}{\partial w}\). $$\frac{\partial x}{\partial u} = 0$$ $$\frac{\partial x}{\partial v} = w$$ $$\frac{\partial x}{\partial w} = v$$ $$\frac{\partial y}{\partial u} = w$$ $$\frac{\partial y}{\partial v} = 0$$ $$\frac{\partial y}{\partial w} = u$$ $$\frac{\partial z}{\partial u} = 2u$$ $$\frac{\partial z}{\partial v} = -2v$$ $$\frac{\partial z}{\partial w} = 0$$

Step 2: Write Out the Jacobian Matrix

Write the 3x3 Jacobian Matrix using the calculated partial derivatives: $$ J(u, v, w) = \begin{bmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} & \frac{\partial x}{\partial w}\\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} & \frac{\partial y}{\partial w}\\ \frac{\partial z}{\partial u} & \frac{\partial z}{\partial v} & \frac{\partial z}{\partial w} \end{bmatrix} = \begin{bmatrix} 0 & w & v\\ w & 0 & u\\ 2u & -2v &0 \end{bmatrix} $$

Step 3: Find Determinant of Jacobian

Calculate the determinant of the Jacobian Matrix to obtain J(u, v, w): $$ J(u, v, w) = \begin{vmatrix} 0 & w & v\\ w & 0 & u\\ 2u & -2v &0 \end{vmatrix} =(0)\cdot((0)(0)-(-2v)(u))-w\cdot(w(0)-(u)(-2v)) + v\cdot(w(-2v)-2u(0)) = 2uvw - (-2uvw) = 4uvw $$ Thus, the Jacobian is \(J(u, v, w) = 4uvw\).