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we known that for each linear operator [itex]\phi:\mathbb{R}^n\rightarrow \mathbb{R}^n[/itex] there exists anadjointoperator [itex]\overline{\phi}[/itex] such that: [tex]<\phi(\mathbf{x}),\mathbf{y}>=<\mathbf{x},\overline{\phi}(\mathbf{y})>[/tex] for allx,yin ℝ^{n}, and where [itex]<\cdot,\cdot>[/itex] is the inner product.

My question is: can we give an analogous definition ofadjointoperator when [itex]\phi:\mathbb{R}^n\rightarrow \mathbb{R}^n[/itex] is adiffeomorphismof ℝ^{n}?

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# Adjoint operators and diffeomorphism

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