Evaluate the force corresponding to the potential energy function \(V(\boldsymbol{r})=c z / r^{3}\), where \(c\) is a constant. Write your answer in vector notation, and also in spherical polars, and verify that it satisfies \(\nabla \wedge \boldsymbol{F}=\mathbf{0}\).

When considering physical systems in classical mechanics, understanding how to derive force from potential energy is a fundamental concept. Potential energy represents a form of energy that is stored within a system due to its position or configuration; for instance, an object at some height above the ground has gravitational potential energy because of its elevated position. The force related to this potential energy can be determined by calculating the negative gradient of the potential energy function.

This method applies widely in physics for different kinds of potentials, such as gravitational, electric, or elastic. In the given exercise, for the potential energy function \(V(\boldsymbol{r}) = \frac{cz}{r^3}\), the force is determined by computing \(-abla {V(\boldsymbol{r})}\). This upwardly points to an important characteristic of conservative forces, where they can be directly derived from the potential energy gradient in this manner. Therefore, deriving the force from potential energy involves calculus and a firm grasp of vector operations.

The gradient is a vector operation that points in the direction of the greatest rate of increase of a scalar field and its magnitude is the steepest slope. When we talk about taking the gradient in Cartesian coordinates, we're referring to finding how a function changes with respect to each of the Cartesian directions (x, y, z).

To calculate the gradient of a scalar field such as potential energy, we take the partial derivative of the function with respect to each of the Cartesian coordinate axes. In the context of our exercise, the potential energy given as \(V(x,y,z) = \frac{cz}{(x^2 + y^2 + z^2)^{3/2}}\), the gradient is computed by taking partial derivatives with respect to \(x\), \(y\), and \(z\), resulting in the force components in Cartesian coordinates. This ability to express the force in Cartesian coordinates is a crucial step for later converting into different coordinate systems, such as spherical polar coordinates.

Spherical polar coordinates provide an alternative to Cartesian coordinates for representing points in three-dimensional space, which can be particularly useful in situations involving spherical symmetry. They are defined by three values: the radial distance \(r\), the polar angle \(\theta\), (measured from the positive z-axis), and the azimuthal angle \(\phi\), (measured from the positive x-axis in the x-y plane).

In the case of our exercise, once we've determined the Cartesian components of the force vector, we can then transform these into spherical polar coordinates using the relations between the systems (\(x = r \times \text{sin}(\theta) \times \text{cos}(\phi)\text{, etc.}\)). After this transformation, the force components are expressed in terms of \(r\), \(\theta\), and \(\phi\), which can be more intuitive depending on the problem's symmetry. The force in spherical coordinates gives insight into how the force behaves radially and angularly within the system.

A crucial concept in classical mechanics is that of a conservative force. A force is said to be conservative if the work done by the force on an object moving from one point to another is independent of the path taken and solely depends on the end points. This is equivalent to stating that the curl of the force field is zero, represented by \(abla \times \boldsymbol{F} = \mathbf{0}\). One important implication of a force being conservative is that it can be derived from a scalar potential energy function, as seen in our original exercise.

Confirming that a force is conservative involves performing vector calculus operations such as calculating the curl. In the exercise, after computing the curl of the force field in spherical coordinates, the result is zero, affirming that the force derived from the given potential energy is indeed conservative. This characteristic has profound implications in physics, indicating, for instance, the conservation of mechanical energy within the system defined by the force.