To find the gradient vector field of the function $f(x, y, z) = 8z e^{-xy}$, we calculate the partial derivatives of $f$ with respect to $x$, $y$, and $z$.

Step-by-Step Solution

1. Partial derivative with respect to $x$: $\frac{\partial f}{\partial x} = \frac{\partial}{\partial x} (8z e^{-xy}) = 8z \cdot (-y) e^{-xy} = -8zy e^{-xy}$

2. Partial derivative with respect to $y$: $\frac{\partial f}{\partial y} = \frac{\partial}{\partial y} (8z e^{-xy}) = 8z \cdot (-x) e^{-xy} = -8zx e^{-xy}$

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

Gradient Vector Field

Now, we can write the gradient vector field $\nabla f$ as: $\nabla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right) = \left( -8zy e^{-xy}, -8zx e^{-xy}, 8 e^{-xy} \right)$

So, the gradient vector field of $f(x, y, z) = 8z e^{-xy}$ is: $\nabla f = \left( -8zy e^{-xy}, -8zx e^{-xy}, 8 e^{-xy} \right)$

Would you like further details on this calculation or have any questions?

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Related Questions

1. How is the gradient vector field used in physics, particularly in fields like electromagnetism?
2. What does the direction of the gradient vector field signify?
3. How would this gradient vector field change if we added a constant to $f(x, y, z)$?
4. Can we interpret this function geometrically, especially in terms of its contours?
5. How does the presence of $e^{-xy}$ affect the behavior of the gradient?

Tip: Remember, the gradient vector points in the direction of the steepest increase of the function.