Answered: Jacobians in three variables Evaluate…

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter6: Linear Transformations

Section6.CR: Review Exercises

Problem 6CR

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Jacobians in three variables Evaluate the Jacobians J(u, ν, w) for the following transformation.

x = ν + w, y = u + w, z = u + ν

Expert Solution

Step 1

We have $x = v + w, y = u + w$ and $z = u + v$ .

Now, we will find partial derivative with respect to each variable.

$\begin{array}{rcl} \frac{\partial x}{\partial u} & = & \frac{\partial}{\partial u} (v + w) \\ = & 0 \end{array}$

$\begin{array}{rcl} \frac{\partial x}{\partial v} & = & \frac{\partial}{\partial v} (v + w) \\ = & 1 \end{array}$

$\begin{array}{rcl} \frac{\partial x}{\partial w} & = & \frac{\partial}{\partial w} (v + w) \\ = & 1 \end{array}$

Step 2

$\begin{array}{rcl} \frac{\partial y}{\partial u} & = & \frac{\partial}{\partial w} (u + w) \\ = & 1 \end{array}$

$\begin{array}{rcl} \frac{\partial y}{\partial v} & = & \frac{\partial}{\partial w} (u + w) \\ = & 0 \end{array}$

$\begin{array}{rcl} \frac{\partial y}{\partial w} & = & \frac{\partial}{\partial w} (u + w) \\ = & 1 \end{array}$

$\begin{array}{rcl} \frac{\partial z}{\partial u} & = & \frac{\partial}{\partial u} (u + v) \\ = & 1 \end{array}$

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Jacobians in three variables Evaluate the Jacobians J(u, ν, w) for the following transformation. x = ν + w, y = u + w, z = u + ν