Verify that each of the following force fields is...

Verify that each of the following force fields is conservative. Then find, for each, a scalar potential $\phi$ such that $\mathbf{F}=-\nabla \phi$.
$$\mathbf{F}=-k \mathbf{r}, \mathbf{r}=\mathbf{i} x+\mathbf{j} y+\mathbf{k} z, \quad k=\mathrm{const.}$$

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Key Concepts

Curl Test

The curl test is a method used to verify whether a force field is conservative. If the curl of the force field is zero over a simply-connected region, then the field is conservative, ensuring that a scalar potential function exists from which the force can be derived.

Conservative Force Field

A conservative force field is one in which the work done by the force along any closed path is zero, implying that the energy transfer is path-independent. This property allows the forces to be expressed as the negative gradient of a scalar potential function, making the analysis of energy conservation and potential energy straightforward in physics.

Scalar Potential

A scalar potential is a single-valued function from which a force field can be derived by taking its negative gradient. In conservative force fields, this function represents the potential energy, and the differences in its values between two points correspond to the work done by the force in moving between those points.

Gradient

The gradient is a vector operator applied to a scalar field that points in the direction of the greatest rate of increase of the function. In the context of force fields, taking the gradient of the scalar potential and then applying a negative sign yields the corresponding force field.

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