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.)
2 Answers
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.
You can derive it for an arbitrary coordinate system.
$\vec{E}= \sum_i E_i \hat{e_i}$
$\nabla \times \vec{E} =\sum_i \nabla(E_i) \times \hat{e_i}+\sum_i E_i (\nabla \times \hat{e_i})$
$\nabla(x_i)=\frac{1}{h_i}\hat{e_i}$
$0= \frac{-\nabla h_i}{h_i^2}\times \hat{e_i} + \frac{1}{h_i}(\nabla \times \hat{e_i})$
$\nabla \times \hat{e_i} = \frac{\nabla h_i}{h_i}\times \hat{e_i}$
$\nabla \times \vec{E} = \sum_i \nabla(E_i) \times \hat{e_i}+ \sum_i E_i \frac{\nabla h_i}{h_i}\times \hat{e_i}$
$\nabla \times \vec{E} = \sum_i \frac{\nabla(h_i E_i)}{h_i}\times \hat{e_i}=\sum_i \nabla(h_iE_i) \times \nabla (x_i)$
$(\nabla \times \vec{E})_a = \sum_i \epsilon_{abc}\frac{1}{h_b}\frac{\partial(h_iE_i)}{\partial x_b}\frac{1}{h_c}\frac{\partial x_i}{\partial x_c}$
$(\nabla \times \vec{E})_a = \sum_i \epsilon_{abc} \frac{1}{h_bh_c}\frac{\partial(h_iE_i)}{\partial x_b}\delta_{ic}$
$(\nabla \times \vec{E})_a= \epsilon_{abc} \frac{1}{h_bh_c}\frac{\partial(h_cE_c)}{\partial x_b}$
$(\nabla \times \vec{E})_a = \epsilon_{abc} \frac{h_a}{h_ah_bh_c}\frac{\partial (h_cE_c)}{\partial x_b}$
For spherical coordinates, $h_r=1, h_\theta=r\sin \phi, h_\phi =r$
$\vec{ds} = dr \hat{r}+ r\sin \phi d\theta \hat{\theta} + r d\phi \hat{\phi}$
$$ \frac{1}{r^2 \sin \phi}\begin{vmatrix} \hat{r} & r\sin \phi \hat{\theta} & r \hat{\phi} \\ \frac{\partial}{\partial r} & \frac{\partial}{\partial \theta} & \frac{\partial }{\partial \phi} \\ E_r & r \sin{\phi} E_\theta & rE_\phi \end{vmatrix}$$
This establishes a general method so for cylindrical coordinates
$\vec{ds} = dr \hat{r} + rd\theta \hat{\theta} + dz \hat{k}$
$$\frac{1}{r} \begin{vmatrix} \hat{r} & r \hat{\theta} & \hat{k} \\ \frac{\partial}{\partial r} &\frac{\partial}{\partial \theta} & \frac{\partial}{\partial z} \\ E_r & rE_\theta & E_z \end{vmatrix} $$
The first row is the line element with the corresponding differentials assumed to be 1, the second row is the partial derivative operators with no scaling factors, the third row is the first row with unit vectors replaced by the components of the vectors being operated on. The coefficient of the matrix is the reciprocal of the product of all the scaling factors.
So in general if $\vec{ds}=h_i dx_i\hat{e_i} +h_j dx_j\hat{e_j} + h_k dx_k \hat{e_k}$
Then $$ \nabla \times \vec{E} = \frac{1}{h_ih_jh_k} \begin{vmatrix} h_i\hat{e_i} & h_j\hat{e_j} & h_k \hat{e_k} \\ \frac{\partial}{\partial x_i } & \frac{\partial}{\partial x_j} & \frac{\partial}{\partial x_k} \\ h_iE_i & h_jE_j & h_k E_k \end{vmatrix} $$
