Problem with Deriving Curl in Spherical Co-ordinates.

Question

The formula for the curl of a vector field $\vec{A}$ in spherical co-ordinates is given as: $$ \begin{align}\nabla \times \vec{A}= \frac{1}{r\sin\theta} \left( \frac{\partial}{\partial \theta} \left(A_\varphi\sin\theta \right) - \frac{\partial A_\theta}{\partial \varphi} \right) &\hat{\mathbf r} \\ {}+ \frac{1}{r} \left( \frac{1}{\sin\theta} \frac{\partial A_r}{\partial \varphi} - \frac{\partial}{\partial r} \left( r A_\varphi \right) \right) &\hat{\boldsymbol \theta} \\ {}+ \frac{1}{r} \left( \frac{\partial}{\partial r} \left( r A_{\theta} \right) - \frac{\partial A_r}{\partial \theta} \right) &\hat{\boldsymbol \varphi}. \end{align}$$ Del in spherical co-ordinates is given as (slide 20): $$\nabla = \frac{\partial}{\partial r} \hat{r}+ \frac1r \frac{\partial}{\partial \theta}\hat{\theta} + \frac1{r\sin{\theta}}\frac{\partial}{\partial \varphi} \hat{\varphi}.$$ Given the above definition of $\nabla$ in spherical co-ordinates, evaluating the curl manually using the determinant gives: $$ \begin{array}{rcl} \nabla \times \vec{A} & = & \begin{vmatrix} \hat{r} & \hat{\theta} & \hat{\varphi} \\ \frac{\partial}{\partial r} & \frac1r \frac{\partial}{\partial \theta} & \frac1{r\sin{\theta}}\frac{\partial}{\partial \varphi} \\ A_r & A_{\theta} & A_{\varphi} \end{vmatrix} \\ & & \begin{align}=\left(\frac1r \frac{\partial}{\partial \theta} A_{\varphi} - \frac1{r\sin{\theta}}\frac{\partial}{\partial \varphi} A_{\theta} \right)\hat{r} \\ +\left(\frac1{r\sin{\theta}} \frac{\partial}{\partial \varphi} A_r - \frac{\partial}{\partial r} A_{\varphi} \right)\hat{\theta} \\ +\left(\frac{\partial}{\partial r} A_{\theta} - \frac1r \frac{\partial}{\partial \theta} A_r \right)\hat{\varphi}. \end{align} \end{array} $$ The two formulae are very similar, however in the top one, various terms have been 'factored' out from inside the partial derivatives (by multiplying the term inside correspondingly), for example the very first term: $$ \frac1{r \sin{\theta}} \frac{\partial}{\partial \theta} A_{\varphi} \sin{\theta} $$ which has been obtained from: $$\frac1r \frac{\partial}{\partial \theta} A_{\varphi}.$$ Is there a mistake in my working, or is there some other reason it is ok to move terms such as $\sin{\theta}$ in and out of the partials freely? (I know that curl is defined fundamentally in terms of integrals etc however I am trying to avoid this explanation.)

Answer 1

According to my sources, the curl should be written as

$$ \nabla \times \mathbf{A} = \frac{1}{r^2\sin\theta}\begin{vmatrix} \hat{r} & r\hat{\theta} & r\sin\theta\hat{\varphi} \\ \frac{\partial}{\partial r} & \frac{\partial}{\partial \theta} & \frac{\partial}{\partial \varphi} \\ A_r & rA_{\theta} & r\sin\theta A_{\varphi} \end{vmatrix} \\ $$

That would resolve your problems.