Problem 40 - Calculus Early Transcendentals

Question

Evaluate the Jacobians \(J(u, v, w)\) for the following transformations. $$u=x-y, v=x-z, w=y+z \quad \text { (Solve for } x, y, \text { and } z \text { first.) }$$

Step-by-Step Solution

Answer

The Jacobian of the given transformation is \(J(u, v, w) = -\frac{1}{2}\).

Step 1: Solve for x, y, and z in terms of u, v, and w

Let's rewrite the given transformations as a set of equations: $$ \begin{cases} u = x - y\\ v = x - z\\ w = y + z \end{cases} $$ Now, we need to solve this system for \(x\), \(y\), and \(z\). To do this, we can first add the first and the third equations to get an expression for \(x\): $$x = \frac{u + w}{2}$$ Similarly, we can subtract the third equation from the first equation to get an expression for \(y\): $$y = \frac{u - w}{2}$$ Now, we can substitute the expression for x we found back into the second equation to get an expression for \(z\): $$z = \frac{v - u}{2}$$

Step 2: Compute Partial Derivatives

Next, we need to compute the partial derivatives of \(x, y, z\) with respect to \(u, v, w\). We will organize these partial derivatives in a 3x3 matrix, called the Jacobian matrix. The matrix looks like this: $$ \begin{pmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} & \frac{\partial x}{\partial w}\\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} & \frac{\partial y}{\partial w}\\ \frac{\partial z}{\partial u} & \frac{\partial z}{\partial v} & \frac{\partial z}{\partial w} \end{pmatrix} $$ We now compute the partial derivatives individually: $$ \begin{aligned} \frac{\partial x}{\partial u} &= \frac{1}{2}, & \frac{\partial x}{\partial v} &= 0, & \frac{\partial x}{\partial w} &= \frac{1}{2}\\ \frac{\partial y}{\partial u} &= \frac{1}{2}, & \frac{\partial y}{\partial v} &= 0, & \frac{\partial y}{\partial w} &= -\frac{1}{2}\\ \frac{\partial z}{\partial u} &= -\frac{1}{2}, & \frac{\partial z}{\partial v} &= \frac{1}{2}, & \frac{\partial z}{\partial w} &= 0 \end{aligned} $$ Filling these partial derivatives in the Jacobian matrix, we get: $$ \begin{pmatrix} \frac{1}{2} & 0 & \frac{1}{2}\\ \frac{1}{2} & 0 & -\frac{1}{2}\\ -\frac{1}{2} & \frac{1}{2} & 0 \end{pmatrix} $$

Step 3: Find the Jacobian

Finally, we need to find the determinant of the Jacobian matrix to get the Jacobian \(J(u, v, w)\): $$ J(u, v, w) = \begin{vmatrix} \frac{1}{2} & 0 & \frac{1}{2}\\ \frac{1}{2} & 0 & -\frac{1}{2}\\ -\frac{1}{2} & \frac{1}{2} & 0 \end{vmatrix} $$ Expanding along the first row, we get: $$ J(u, v, w) = \frac{1}{2}\begin{vmatrix} 0 & -\frac{1}{2}\\ \frac{1}{2} & 0 \end{vmatrix} - 0\begin{vmatrix} \frac{1}{2} & -\frac{1}{2}\\ -\frac{1}{2} & 0 \end{vmatrix} + \frac{1}{2}\begin{vmatrix} \frac{1}{2} & 0\\ -\frac{1}{2} & \frac{1}{2} \end{vmatrix} $$ Calculating the determinants inside, we find: $$ J(u, v, w) = \frac{1}{2}(-\frac{1}{4}) + 0 - \frac{1}{2}(\frac{1}{4}) = -\frac{1}{4} - \frac{1}{4} = -\frac{1}{2} $$ Thus, the Jacobian of the given transformation is \(J(u, v, w) = -\frac{1}{2}\).