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

$\mathbf{F} = (y + z)\hat{i} + (x + 3)\hat{j} + (x + y)\hat{k}.$

Since the force field is conservative, 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:

$\frac{\partial \phi}{\partial x} = y + z,$ $\frac{\partial \phi}{\partial y} = x + 3,$ $\frac{\partial \phi}{\partial z} = x + y.$

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:

$\frac{\partial \phi}{\partial x} = y + z.$

Integrating with respect to $x$:

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

where $g(y, z)$ is an arbitrary function of $y$ and $z$.

This simplifies to:

$\phi(x, y, z) = xy + xz + g(y, z).$

Step 2: Differentiate with respect to $y$

Next, use the second equation:

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

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

$x + \frac{\partial g(y, z)}{\partial y} = x + 3.$

This implies:

$\frac{\partial g(y, z)}{\partial y} = 3.$

Integrating with respect to $y$:

$g(y, z) = 3y + h(z),$

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

Thus, we have:

$\phi(x, y, z) = xy + xz + 3y + h(z).$

Step 3: Differentiate with respect to $z$

Finally, use the third equation:

$\frac{\partial \phi}{\partial z} = \frac{\partial}{\partial z} \left(xy + xz + 3y + h(z)\right) = x + \frac{\partial h(z)}{\partial z}.$

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

$x + \frac{\partial h(z)}{\partial z} = x + y.$

This implies:

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

Integrating with respect to $z$:

$h(z) = yz + C,$

where $C$ is a constant.

Final Scalar Potential

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

$\phi(x, y, z) = xy + xz + 3y + yz + C.$

This can be further simplified to:

$\phi(x, y, z) = xy + xz + yz + 3y + C,$

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 check if a given force field is conservative?
2. What is the relationship between the scalar potential and potential energy in a physical system?
3. How would the scalar potential change if the force field had an additional constant term?
4. How can you find the work done by this force field along a specific path?
5. What are some examples of physical situations where a conservative force field like this one would appear?

Tip: When dealing with conservative force fields, always ensure that the curl of the force field is zero, as this is a key property of conservative forces.