Is $Cr^a\hat r$ always a conservative vector field?

Question

Is the vector field $\vec r(r, \theta, \phi) = Cr^a\hat r$, where $C, a\in \Bbb R$ are constants and $r \in \Bbb R^+ \cup \{0\}$ is the radial component, always a conservative vector field? I really just want to know if I have a force field of this type can I always find a potential function?

Answer 1

Yes. The curl in spherical polars involves $\theta$ and $\phi$ derivatives of the radial component, but not $r$ derivatives.

$$\nabla\times{\bf u}=\hat{\bf r}\frac1{r\sin\theta}\left[\frac\partial{\partial\theta}(u_\phi\sin\theta)-\frac{\partial u_\theta}{\partial\phi}\right]+\hat{\boldsymbol{\theta}}\frac1r\left[\frac1{\sin\theta}\frac{\partial u_r}{\partial\phi}-\frac\partial{\partial r}(ru_\phi)\right]+\hat{\boldsymbol{\phi}}\frac1r\left[\frac\partial{\partial r}(ru_\theta)-\frac{\partial u_r}{\partial\theta}\right].$$

Hence, $\nabla\times{\bf r}= 0$ and therefore the vector field is conservative.

As the grad of a scalar field in polar coordinates is given by

$$\nabla f=\hat{\bf r}\frac{\partial f}{\partial r}+\hat{\boldsymbol{\theta}}\frac1r\frac{\partial f}{\partial\theta}+\hat{\boldsymbol{\phi}}\frac1{r\sin\theta}\frac{\partial f}{\partial\phi},$$ it is easy to integrate your vector field to find a corresponding potential scalar field.