Finding the curl - what am I doing wrong?

Question

I must be making a really daft mistake. If $\mathbf{F}=-y\mathbf{e}_x+x\mathbf{e}_y=r\mathbf{e}_{\phi}$ (in Cartesian and Cylindrical polars respectively):

$$\nabla\times\mathbf{F}=\begin{pmatrix}\partial/\partial x\\ \partial/\partial y\\\partial/\partial z\end{pmatrix}\times \begin{pmatrix}-y\\x\\0\end{pmatrix}=2\mathbf{e}_z$$

$$\nabla\times\mathbf{F}=\begin{pmatrix}\partial/\partial r\\ \tfrac1r\partial/\partial \phi\\\partial/\partial z\end{pmatrix}\times \begin{pmatrix}0\\r\\0\end{pmatrix}=\mathbf{e}_z$$

Where is the mistake?

are you sure that is how you calculate the curl in cylindrical coordinates? — James S. Cook, Mar 01 '14 at 16:51

score 2 · Accepted Answer · answered Mar 01 '14 at 16:51

2

Curls do not work the way you think they do for curvilinear coordinates. See here for the actual formulae.

answered Mar 01 '14 at 16:51

Cameron Williams

29,432

Finding the curl - what am I doing wrong?

1 Answers1