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$U,V$ vector fields on $\mathbb{R}^3$ $$U = \frac{\partial}{\partial x},V = F\frac{\partial}{\partial y}+G\frac{\partial}{\partial z}$$ answer key says: $$[U,V] = F_x\frac{\partial}{\partial y}+G_x\frac{\partial}{\partial z} $$ By my calculations, given $f:\mathbb{R}^3\rightarrow\mathbb{R}$ smooth \begin{align} [U,V]f & = U(V( f ))-V(U( f ))\\ & = U\left(F\frac{\partial f }{\partial y}+G\frac{\partial f }{\partial z}\right)+V\left(\frac{\partial f }{\partial x}\right)\\ & = \left(\frac{\partial F}{\partial x}\frac{\partial f}{\partial y}+\frac{\partial G}{\partial x}\frac{\partial f}{\partial z}\right)+\left(F\frac{\partial^2 f}{\partial y\partial x} + G\frac{\partial^2 f}{\partial z\partial x}\right) \end{align}
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Sun, 17 Dec 2017 04:23 GMT