To determine which vector fields among $\mathbf{F}$, $\mathbf{G}$, and $\mathbf{H}$ are conservative, we need to analyze whether each field has a potential function. A conservative vector field is typically one where the curl of the field is zero in a simply connected domain.

Let's examine each vector field:

1. Vector Field $\mathbf{F} = \langle 2, 0, 1 \rangle$:

   • $\mathbf{F}$ is a constant vector field, meaning that the components do not depend on $x$, $y$, or $z$.
   • The curl of a constant field is zero because all partial derivatives are zero.
   • Therefore, $\mathbf{F}$ is conservative.

2. Vector Field $\mathbf{G} = \langle x + y, y + z, x + z \rangle$:

   • To check if $\mathbf{G}$ is conservative, we calculate the curl: $\nabla \times \mathbf{G} = \left( \frac{\partial}{\partial y}(x + z) - \frac{\partial}{\partial z}(y + z), \frac{\partial}{\partial z}(x + y) - \frac{\partial}{\partial x}(x + z), \frac{\partial}{\partial x}(y + z) - \frac{\partial}{\partial y}(x + y) \right)$
   • After computing the partial derivatives, we find that the curl is not zero, indicating that $\mathbf{G}$ is not conservative.

3. Vector Field $\mathbf{H} = \langle 2xy, x^2 + z^2, 2yz \rangle$:

   • To check if $\mathbf{H}$ is conservative, we calculate the curl: $\nabla \times \mathbf{H} = \left( \frac{\partial}{\partial y}(2yz) - \frac{\partial}{\partial z}(x^2 + z^2), \frac{\partial}{\partial z}(2xy) - \frac{\partial}{\partial x}(2yz), \frac{\partial}{\partial x}(x^2 + z^2) - \frac{\partial}{\partial y}(2xy) \right)$
   • After computing these partial derivatives, we also find that the curl is not zero, meaning that $\mathbf{H}$ is not conservative.

Conclusion:

Only $\mathbf{F}$ is a conservative vector field because it is the only one with a zero curl. Therefore, the correct answer is:

• $\mathbf{F}$ only

Would you like further details on the calculations for the curl of each vector field?

Here are five related questions for further exploration:

1. What are the conditions for a vector field to be conservative in a three-dimensional space?
2. How is the potential function of a conservative field derived if it exists?
3. What are some physical examples of conservative vector fields?
4. How does the concept of a conservative field apply in different coordinate systems?
5. Can a non-conservative field be made conservative by restricting the domain?

Tip: For a vector field in three dimensions, checking if the curl is zero is a quick test to determine if the field is conservative, provided the domain is simply connected.