I have a vector wavefield in the polar coordinate system $ \boldsymbol{u} = u_{r} \hat{r} + u_{\theta} \hat{\theta}$. At a point ($r, \theta_{1}$), I want to calculate gradient or derivative of the wavefield in a different direction ($\theta_{2}$) such that the rotation angle between the two directions is $\theta_{3} = \theta_{1} - \theta_{2}$. Are there any transformation or calculus formulae that can accomplish this is the polar coordinate system ?
I was able to do this by transforming to Cartesian coordinate system $ \boldsymbol{u} = u_{x} \hat{x} + u_{y} \hat{y}$, $r=\sqrt{x^{2}+y^{2}}$, $\cos{\theta}=\frac{x}{\sqrt{x^{2}+y^{2}}}$ and $\sin{\theta}=\frac{y}{\sqrt{x^{2}+y^{2}}}$. For wavefield $ \boldsymbol{u} = u_{x} \hat{x} + u_{y} \hat{y}$, the axial gradient in any direction $\hat{\theta_{2}}$ is $e_{\hat{\theta_{2}\theta_{2}}} = \cos^{2}{\theta_{2}} \frac{\partial u_x}{\partial x} + \sin^{2}{\theta_{2}} \frac{\partial u_y}{\partial y} + (\cos{\theta_{2}})(\sin{\theta_{2}})(\frac{\partial u_x}{\partial y} + \frac{\partial u_y}{\partial x}) $ and then I was able to transform the expression back to the polar coordinate system. The final results depends only on the rotation angle between the two directions $\theta_{3}$ and $r$. I was wondering if there are standard formula to calculate the gradient of a vector field in the polar coordinate system in any arbitrary direction without going through the transformation to and fro the cartesian coordinate system.
