We are asked to find the gradient $\nabla f$ of the function $f(x, y, z) = x^2y + y^2z + z^2x$.

1. Problem Description

We are asked to find the gradient $\nabla f$ of the function $f(x, y, z) = x^2y + y^2z + z^2x$.

2. Solution Steps

The gradient $\nabla f$ of a scalar function $f(x, y, z)$ is a vector given by:

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

We need to compute the partial derivatives of $f(x, y, z) = x^2y + y^2z + z^2x$ with respect to $x$, $y$, and $z$.

$$\frac{\partial f}{\partial x} = \frac{\partial}{\partial x} (x^2y + y^2z + z^2x) = 2xy + 0 + z^2 = 2xy + z^2$$

$$\frac{\partial f}{\partial y} = \frac{\partial}{\partial y} (x^2y + y^2z + z^2x) = x^2 + 2yz + 0 = x^2 + 2yz$$

$$\frac{\partial f}{\partial z} = \frac{\partial}{\partial z} (x^2y + y^2z + z^2x) = 0 + y^2 + 2zx = y^2 + 2zx$$

Therefore, the gradient is:

$$\nabla f = (2xy + z^2, x^2 + 2yz, y^2 + 2zx)$$

3. Final Answer

$\nabla f = (2xy + z^2, x^2 + 2yz, y^2 + 2zx)$