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The problem asks to find the **Jacobian matrix** \( J \left( \frac{u, v, w}{x, y, z} \right) \) for the system of equations provided. The Jacobian matrix represents the partial derivatives of the functions \( u \), \( v \), and \( w \) with respect to \( x \), \( y \), and \( z \).

### Given equations:
1. \( u = x + y + z \)
2. \( u^2 v = y + z \)
3. \( u^3 w = z \)

### Step-by-step approach to solving for the Jacobian:
The Jacobian matrix \( J \) is a \( 3 \times 3 \) matrix of the partial derivatives of \( u \), \( v \), and \( w \) with respect to \( x \), \( y \), and \( z \). The general form of the Jacobian is:

\[
J = \begin{pmatrix}
\frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} & \frac{\partial u}{\partial z} \\
\frac{\partial v}{\partial x} & \frac{\partial v}{\partial y} & \frac{\partial v}{\partial z} \\
\frac{\partial w}{\partial x} & \frac{\partial w}{\partial y} & \frac{\partial w}{\partial z}
\end{pmatrix}
\]

1. **Partial derivatives for \( u = x + y + z \):**
   \[
   \frac{\partial u}{\partial x} = 1, \quad \frac{\partial u}{\partial y} = 1, \quad \frac{\partial u}{\partial z} = 1
   \]

2. **For the equation \( u^2 v = y + z \):**
   Differentiate this equation to find partial derivatives of \( v \). First, rearrange for \( v \):
   \[
   v = \frac{y + z}{u^2}
   \]
   Now compute the partial derivatives:
   \[
   \frac{\partial v}{\partial x} = -\frac{2(y + z)}{u^3}, \quad \frac{\partial v}{\partial y} = \frac{1}{u^2}, \quad \frac{\partial v}{\partial z} = \frac{1}{u^2}
   \]

3. **For the equation \( u^3 w = z \):**
   Rearrange for \( w \):
   \[
   w = \frac{z}{u^3}
   \]
   Now compute the partial derivatives:
   \[
   \frac{\partial w}{\partial x} = -\frac{3z}{u^4}, \quad \frac{\partial w}{\partial y} = -\frac{3z}{u^4}, \quad \frac{\partial w}{\partial z} = \frac{1}{u^3}
   \]

### Jacobian Matrix:
\[
J = \begin{pmatrix}
1 & 1 & 1 \\
-\frac{2(y + z)}{u^3} & \frac{1}{u^2} & \frac{1}{u^2} \\
-\frac{3z}{u^4} & -\frac{3z}{u^4} & \frac{1}{u^3}
\end{pmatrix}
\]

This is the Jacobian matrix for the system of equations.

Would you like further clarification on any step? 

Here are some related questions for better understanding:
1. What is the significance of the Jacobian matrix in multivariable calculus?
2. How can the Jacobian be used to transform coordinates in a multivariable system?
3. Can you explain how the chain rule is applied when deriving Jacobians?
4. What are the conditions for a Jacobian to be invertible?
5. How is the Jacobian used in optimization problems?

**Tip**: The Jacobian is often used in nonlinear systems to approximate small changes in multivariable functions, so it's very useful in numerical methods and differential equations.

Find the Jacobian matrix for the given system of equations: u = x + y + z, u^2v = y + z, u^3w = z.

Explore the calculation of the Jacobian matrix for the system of equations u = x + y + z, u^2v = y + z, and u^3w = z. This example covers key concepts in multivariable calculus, including partial derivatives and Jacobian matrices. Suitable for university-level students.

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