To find the conservative vector field for the given potential function $f(x, y, z) = 8xyz$, we calculate its gradient, $\nabla f$. The gradient is defined as:

$\nabla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right)$

Step 1: Partial derivatives

1. Partial derivative with respect to $x$: $\frac{\partial f}{\partial x} = \frac{\partial}{\partial x}(8xyz) = 8yz$

2. Partial derivative with respect to $y$: $\frac{\partial f}{\partial y} = \frac{\partial}{\partial y}(8xyz) = 8xz$

3. Partial derivative with respect to $z$: $\frac{\partial f}{\partial z} = \frac{\partial}{\partial z}(8xyz) = 8xy$

Step 2: Combine results

The conservative vector field is: $\nabla f = (8yz, 8xz, 8xy)$

This is the required vector field.

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Related Questions

1. What are conservative vector fields, and how are they used in physics?
2. How can you verify if a given vector field is conservative?
3. Can a potential function exist for non-conservative fields?
4. How does the gradient operation relate to divergence and curl?
5. What is the significance of conservative fields in fluid dynamics?

Tip

Always check your calculations for each partial derivative step-by-step to avoid errors, especially for complex potential functions.