I have a quantity $[UV]$ where the $[\hspace{5pt}]$ represents ensemble averaging and $U, V$ are the velocity components in $X$ and $Y$ directions. In a new reference frame (say: $x, y$), which is rotated by an angle "$a$" in the CLOCKWISE direction with respect to the reference frame $(X, Y)$, lets say the velocity components are $(u,v)$. What would be the product quantity $[uv]$ in the new coordinate system? I applied the Jacobian rotation and got the following: $$[uv] = 0.5([UU] - [VV])\sin2a + [UV]\cos2a$$. Can anybody please confirm?
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