How does taking a functional derivative affect the units of the functional?
As an example, considering $\mathcal{F} = \mathcal{F}[h,\nabla h]$ where $\mathcal{F}$ is the free energy functional of some fluid system and $h$ represents the film height. Then in fundamental units we would have: $[\mathcal{F}]=\frac{ML^2}{T^2}$. My question is what would the units of the variational derivative be: $[\frac{\delta \mathcal{F}}{\delta h}]$? Is it the same as taking a regular or partial derivative of a function? i.e. would it just have the units of energy over length?