# How to define the grad, div and the curl of a distribution?

I have learned how to define the derivative or partial derivative of a distribution, but I still can't find a clear definition of the grad, div and curl of a distribution, I would appreciate it if you could write down the detailed definition of the definitions of these operators of distribution or recommend me some books about the curl, div and grad of distributions

• These operators are defined as linear combinations of partial derivatives (at least in $\mathbb{R}^n$), so one can take this as a definition also in case of distributions. Commented May 6, 2021 at 15:26
• Forget distributions for a moment and ask yourself what the dual of each of these is just on regular smooth functions. $\operatorname{grad}$ maps scalar-valued functions to vector-valued functions. So the dual of $\operatorname{grad}$ would have to take vector-valued functions to scalar-valued. Of course if you want to forget for a moment that distributions should take scalar-valued functions to the scalar field (and thus you run into issues above), you can do what @MichałMiśkiewicz suggests. Commented May 6, 2021 at 15:26
• I have figured out the definition of grad and div, but I don't know how to express the curl, could you elaborate on it? thank you @Cameron Williams Commented May 6, 2021 at 16:21