The problem asks to find the scalar potential $\phi(x, y, z)$ for the conservative force field given by:

$\mathbf{F} = (yz + 2x)\hat{i} + (xz + 2y)\hat{j} + (xy + 2z)\hat{k}.$

Since the force field is conservative, we know that there exists a scalar potential function $\phi(x, y, z)$ such that:

$\mathbf{F} = \nabla \phi = \left(\frac{\partial \phi}{\partial x}\right)\hat{i} + \left(\frac{\partial \phi}{\partial y}\right)\hat{j} + \left(\frac{\partial \phi}{\partial z}\right)\hat{k}.$

This implies:

$\frac{\partial \phi}{\partial x} = yz + 2x,$ $\frac{\partial \phi}{\partial y} = xz + 2y,$ $\frac{\partial \phi}{\partial z} = xy + 2z.$

We will integrate these partial derivatives step by step to find $\phi(x, y, z)$.

Step 1: Integrate with respect to $x$

Start with the first equation:

$\frac{\partial \phi}{\partial x} = yz + 2x.$

Integrating with respect to $x$:

$\phi(x, y, z) = x(yz + 2x) + g(y, z),$

where $g(y, z)$ is an arbitrary function of $y$ and $z$ (since the derivative of $g(y, z)$ with respect to $x$ is zero).

Simplifying, we get:

$\phi(x, y, z) = xyz + x^2 + g(y, z).$

Step 2: Differentiate with respect to $y$

Next, use the second equation:

$\frac{\partial \phi}{\partial y} = \frac{\partial}{\partial y} \left(xyz + x^2 + g(y, z)\right) = xz + \frac{\partial g(y, z)}{\partial y}.$

Set this equal to the given expression for $\frac{\partial \phi}{\partial y}$:

$xz + \frac{\partial g(y, z)}{\partial y} = xz + 2y.$

This implies:

$\frac{\partial g(y, z)}{\partial y} = 2y.$

Integrating with respect to $y$:

$g(y, z) = y^2 + h(z),$

where $h(z)$ is an arbitrary function of $z$.

Thus, we have:

$\phi(x, y, z) = xyz + x^2 + y^2 + h(z).$

Step 3: Differentiate with respect to $z$

Finally, use the third equation:

$\frac{\partial \phi}{\partial z} = \frac{\partial}{\partial z} \left(xyz + x^2 + y^2 + h(z)\right) = xy + \frac{\partial h(z)}{\partial z}.$

Set this equal to the given expression for $\frac{\partial \phi}{\partial z}$:

$xy + \frac{\partial h(z)}{\partial z} = xy + 2z.$

This implies:

$\frac{\partial h(z)}{\partial z} = 2z.$

Integrating with respect to $z$:

$h(z) = z^2 + C,$

where $C$ is a constant.

Final Scalar Potential

Thus, the scalar potential $\phi(x, y, z)$ is:

$\phi(x, y, z) = xyz + x^2 + y^2 + z^2 + C,$

where $C$ is an arbitrary constant.

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Would you like more details or have any questions? Here are five related questions you might find interesting:

1. How can you verify that the given force field is conservative?
2. What is the physical significance of the scalar potential in this context?
3. How would you find the work done by the force field over a path between two points?
4. Can a non-conservative force field have a scalar potential? Why or why not?
5. What would the scalar potential look like if the force field had a different set of components?

Tip: In physics, conservative force fields are often associated with potential energy functions, which are scalar potentials that describe the energy stored in a system due to its position.