What is the gradient in polar coordinates?

[SeM]

Hi, on this page: https://en.wikipedia.org/wiki/Laplace_operator#Two_dimensions

the Laplacian is given for polar coordinates, however this is only for the second order derivative, also described as \delta f . Can someone point me to how to represent the first-order Laplacian operator in polar coordinates? I found this at https://math.stackexchange.com/questions/586848/how-to-obtain-the-gradient-in-polar-coordinates, however, I am not sure its correct:

\nabla = \boldsymbol{e_r} \frac{\partial}{\partial r}+ \boldsymbol{e_{\theta}} \frac 1r \frac{\partial}{\partial \theta}.

Thanks!

 

[Orodruin]

the first-order Laplacian operator

The Laplacian by definition is a second order differential operator. It is defined as the divergence of the gradient of a scalar field. There is no such thing as a first order Laplacian.

There are three first order differential operators you will encounter in basic vector analysis, the gradient, the divergence and the curl. You will often see them written as $\nabla$, $\nabla \cdot$, and $\nabla \times$. However, you should be very careful here. The reason for expressing them like this is that it becomes correct to schematically write $\nabla = \vec e_1 \partial_1 + \vec e_2 \partial_2 + \vec e_3 \partial_3$ and apply the $\cdot$ and $\times$ as scalar and cross products with the derivatives acting on what is to the right. This only works in Cartesian coordinates. In curvilinear coordinates you will have to find the appropriate expressions and they will not involve writing $\nabla$ as a vector and blindly applying the scalar and cross products.

See https://en.wikipedia.org/wiki/Curvilinear_coordinates#Differentiation for the correct expressions in general coordinates. This is also something you should be able to find in any introductory textbook on vector analysis.

Please also be aware that the "A" tag you put on this thread would imply that you have a graduate level understanding of the subject and expect an answer at that level. Based on the content of the question, the thread level is at most an "I" and I have changed it accordingly.

 

[DrGreg]

\nabla = \boldsymbol{e_r} \frac{\partial}{\partial r}+ \boldsymbol{e_{\theta}} \frac 1r \frac{\partial}{\partial \theta}.

By the way, this forum supports $\LaTeX$ but you need to put double-$ signs before and after the code:$$
\nabla = \boldsymbol{e_r} \frac{\partial}{\partial r}+ \boldsymbol{e_{\theta}} \frac 1r \frac{\partial}{\partial \theta}.
$$

 

[SeM]

Thanks!