To solve this problem, let's begin by breaking down the given functions and finding the required partial derivatives. We have:

1. Function Definitions: $w = -x^2 + 2y^2 + 2z^2$ $z = -x^2 - 3y^2$

2. Objective:

   • We are asked to find:
     • $\left( \frac{\partial w}{\partial y} \right)_z$
     • $\left( \frac{\partial w}{\partial z} \right)_x$
     • $\left( \frac{\partial w}{\partial z} \right)_y$

Step-by-Step Solution:

Part (a): $\left( \frac{\partial w}{\partial y} \right)_z$

To find this partial derivative, we need to express $w$ as a function of $y$ alone, with $z$ treated as a constant. Since $z = -x^2 - 3y^2$, we differentiate $w$ with respect to $y$ while holding $z$ constant.

1. Rewrite $w$ in terms of $y$: $w = -x^2 + 2y^2 + 2z^2$
2. Differentiate $w$ with respect to $y$, treating $z$ as a constant: $\frac{\partial w}{\partial y} = 4y$

Thus, $\left( \frac{\partial w}{\partial y} \right)_z = 4y$

Part (b): $\left( \frac{\partial w}{\partial z} \right)_x$

To find this partial derivative, we take the derivative of $w$ with respect to $z$ while treating $x$ as constant.

1. Differentiate $w$ with respect to $z$: $\frac{\partial w}{\partial z} = 4z$

Thus, $\left( \frac{\partial w}{\partial z} \right)_x = 4z$

Part (c): $\left( \frac{\partial w}{\partial z} \right)_y$

For this partial derivative, we need to differentiate $w$ with respect to $z$ while treating $y$ as constant.

Since $w$ is already expressed in terms of $z$ independently of $y$, the partial derivative remains:

$\left( \frac{\partial w}{\partial z} \right)_y = 4z$

Summary of Answers:

• (a) $\left( \frac{\partial w}{\partial y} \right)_z = 4y$
• (b) $\left( \frac{\partial w}{\partial z} \right)_x = 4z$
• (c) $\left( \frac{\partial w}{\partial z} \right)_y = 4z$

Would you like further details on the steps or have any questions about the process?

Follow-Up Questions

1. What would the partial derivative $\left( \frac{\partial w}{\partial x} \right)_y$ be?
2. How would the answer change if $z$ were defined as a function involving $x$ and $y$ differently?
3. What happens if we substitute the expression for $z$ into $w$ before differentiating?
4. Can we use the chain rule to find higher-order derivatives of $w$ with respect to $y$ and $z$?
5. How do these partial derivatives relate to the concept of gradient in multivariable calculus?

Tip

When working with partial derivatives where multiple variables are related, isolating each variable’s dependency and treating constants accordingly is key to simplifying the process.