To calculate the curl in spherical coordinates, we use the formula for the curl of a vector field in a curvilinear coordinate system. In spherical coordinates $(r, \theta, \phi)$, the curl of a vector field $\mathbf{A} = A_r \hat{r} + A_\theta \hat{\theta} + A_\phi \hat{\phi}$ is given by the following expression:

Curl in Spherical Coordinates

\begin{vmatrix} \hat{r} & r \hat{\theta} & r \sin\theta \hat{\phi} \\ \frac{\partial}{\partial r} & \frac{\partial}{\partial \theta} & \frac{\partial}{\partial \phi} \\ A_r & r A_\theta & r \sin\theta A_\phi \end{vmatrix}$$ ### Expanding the Determinant This formula involves a **3x3 determinant**. Breaking it down for each component: #### Radial Component: $$(\nabla \times \mathbf{A})_r = \frac{1}{r \sin\theta} \left( \frac{\partial}{\partial \theta}(r \sin\theta A_\phi) - \frac{\partial}{\partial \phi}(r A_\theta) \right)$$ #### Polar Component: $$(\nabla \times \mathbf{A})_\theta = \frac{1}{r} \left( \frac{\partial}{\partial \phi}(A_r) - \frac{\partial}{\partial r}(r \sin\theta A_\phi) \right)$$ #### Azimuthal Component: $$(\nabla \times \mathbf{A})_\phi = \frac{1}{r} \left( \frac{\partial}{\partial r}(r A_\theta) - \frac{\partial}{\partial \theta}(A_r) \right)$$ ### Final Form of the Curl in Spherical Coordinates: Thus, the curl $$\nabla \times \mathbf{A}$$ in spherical coordinates is: $$\nabla \times \mathbf{A} = \left( \frac{1}{r \sin\theta} \left( \frac{\partial}{\partial \theta}(r \sin\theta A_\phi) - \frac{\partial}{\partial \phi}(r A_\theta) \right) \right) \hat{r} + \left( \frac{1}{r} \left( \frac{\partial}{\partial \phi}(A_r) - \frac{\partial}{\partial r}(r \sin\theta A_\phi) \right) \right) \hat{\theta} + \left( \frac{1}{r} \left( \frac{\partial}{\partial r}(r A_\theta) - \frac{\partial}{\partial \theta}(A_r) \right) \right) \hat{\phi}$$ ### Step-by-Step Calculation: 1. **Identify the components** of your vector field $$\mathbf{A}$$, i.e., $$A_r$$, $$A_\theta$$, and $$A_\phi$$. 2. **Apply the partial derivatives** as required for each component of the curl. 3. **Simplify** the resulting expressions. --- Would you like to see an example of calculating the curl in spherical coordinates with a specific vector field? --- ### Related Questions: 1. How do I calculate divergence in spherical coordinates? 2. What is the physical interpretation of the curl in spherical coordinates? 3. How do I apply the curl formula to a specific vector field? 4. How do the coordinate transformations affect the curl? 5. What are some common mistakes when calculating curl in spherical coordinates? ### Tip: When calculating curl in spherical coordinates, make sure to correctly handle the $$r \sin\theta$$ factor in both the determinant and the derivatives.