Suppose I have a functional $$ E=\int F(y_{1,1},..y_{1,n},y_{2,1}\ldots,y_{n,n})d\boldsymbol{x}\,, $$ where $\boldsymbol{y}:\mathbb{R}^{n}\to\mathbb{R}^{n},\,\boldsymbol{y}(\boldsymbol{x})=\left(y_{1}(x_{1},...x_{n}),...,y_{n}(x_{1},...x_{n})\right)$, and $y_{1,1},...,y_{n,n}$ are partial derivatives, i.e. $y_{i,j}=\dfrac{dy_{i}}{dx_{i}}$.
How the variation of the functional $\dfrac{\delta E(\boldsymbol{y})}{\delta(\boldsymbol{y})}$ is changed under a rigid coordinates transformation (of both the domain and the image.) $$ \left(x_{1},...,x_{n}\right)^{T}=V\left(v_{1},...,v_{n}\right)^{T};\,\left(y_{1},...,y_{n}\right)^{T}=U\left(u_{1},...,u_{n}\right)^{T}, $$
where $U,V$ are orthogonal $n\times n$ matrices.
In other words, given $\dfrac{\delta E(\boldsymbol{y(x)})}{\delta(\boldsymbol{y(x)})}$ what would be the expression for $\dfrac{\delta E(\boldsymbol{u(v)})}{\delta(\boldsymbol{u(v)})}$?
Any comments would be appreciated!