# Change of variation under rigid coordinate transformation

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!

• P.S. To be sure that there is no ambiguity in my question; by "variation'' I mean the one that follows from the Euler Lagrange equation for $E$ that depends on partial derivatives of $n$-functions of $n$-variables i.e., $$\dfrac{\delta E(\boldsymbol{y(x)})}{\delta(\boldsymbol{y(x)})}=\left(\underset{i=1}{\overset{n}{\sum}}\dfrac{\partial}{\partial x_{i}}\dfrac{\partial F}{\partial y_{1,i}},\ldots,\underset{i=1}{\overset{n}{\sum}}\dfrac{\partial}{\partial x_{i}}\dfrac{\partial F}{\partial y_{n,i}}\right)^{T}.$$ Sep 7 '16 at 16:10