Find which of the following forces are conservative, and for those that are find the corresponding potential energy function ( \(a\) and \(b\) are constants, and \(\boldsymbol{a}\) is a constant vector): (a) \(F_{x}=a x+b y^{2}, \quad F_{y}=a z+2 b x y, \quad F_{z}=a y+b z^{2}\); (b) \(F_{x}=a y, \quad F_{y}=a z, \quad F_{z}=a x\); (c) \(F_{r}=2 a r \sin \theta \sin \varphi, \quad F_{\theta}=\operatorname{arcos} \theta \sin \varphi, \quad F_{\varphi}=\operatorname{arcos} \varphi\); (d) \(\boldsymbol{F}=\boldsymbol{a} \wedge \boldsymbol{r}\); (e) \(\boldsymbol{F}=r \boldsymbol{a} ;\) (f) \(\boldsymbol{F}=\boldsymbol{a}(\boldsymbol{a} \cdot \boldsymbol{r}) .\)

In physics, potential energy refers to the energy stored within a system due to the position of objects in a force field. For example, an object held at a certain height above the ground has gravitational potential energy due to Earth's gravitational field. In the context of conservative forces, potential energy plays a crucial role. A conservative force is one where the work done in moving an object between two points is independent of the path taken, only depending on the endpoints.

For conservative forces, such as gravity and electrostatic forces, there exists a scalar potential energy function, often denoted as U, such that the negative gradient of U with respect to position gives the force vector. Mathematically, we express this relation as:$$ \boldsymbol{F} = -abla U(\boldsymbol{r}) $$
The work done by a conservative force in moving an object from point A to point B is equal to the decrease in the potential energy of the system. Therefore in the solution example provided, in step (a), after verifying the force is conservative, we integrated the force components to obtain the potential energy function, encapsulating all the energy due to the positioning within the force field.

The curl of a vector field is a vector operator that describes the infinitesimal rotation of the field in three-dimensional space. In essence, the curl provides us with a way to measure the 'twisting' or the rotational tendency at any point within the field. For a force field \( \boldsymbol{F} \), we denote the curl as \( abla \times \boldsymbol{F} \). It is defined as:$$ (\frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z}, \frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x}, \frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y}) $$

If the curl of a force field is equal to the zero vector everywhere, it indicates that there is no 'circulation' or rotational effect, and such a force is labeled as conservative. The significance of this property means that the path integral of a conservative force field around any closed loop is zero. In our exercise, steps 1, 3, 5, and 7 calculate the curl to determine the conservativeness of forces. For instance, in step 3 for force (b), the nonzero curl implies the force is not conservative and thus lacks an associated potential energy function.

The gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field and whose magnitude is the rate of that increase. Mathematically, for a scalar field U(x, y, z), the gradient is given by:$$ abla U = (\frac{\partial U}{\partial x}, \frac{\partial U}{\partial y}, \frac{\partial U}{\partial z}) $$
It provides a way to understand how a quantity, such as potential energy, changes spatially. In our textbook solution, the integration of gradient equations arises after ensuring that the forces considered are conservative. From steps 2 and 8, we see that we obtain the scalar potential energy function U(x, y, z) by integrating the components of the conservative force field, essentially reversing the gradient operation. This process is fundamental in classical mechanics, allowing us to link potential energy with the forces acting within a system.