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given a system of curvilinear coordinates [itex]x_i=x_i(u_1,\ldots,u_n)[/itex]; [itex]u_i=u_i(x_1,\ldots,x_n)[/itex] and considering the position vector [itex]\mathbf{r}=x_1\mathbf{e}_1+\ldots+x_n\mathbf{e}_n[/itex] there is the well-known identity that defines thereciprocal frame:

[tex]\frac{\partial \mathbf{r}}{\partial u_i }\cdot \nabla u_j = \delta^i_j[/tex]

I tried to verify it by myself but I cannot see where is the mistake:

[tex]\frac{\partial \mathbf{r}}{\partial u_i }\cdot \nabla u_j=[/tex]

[tex]=(\frac{\partial x_1}{\partial u_i }\mathbf{e}_1+ \ldots + \frac{\partial x_n}{\partial u_i }\mathbf{e}_n )\cdot (\frac{\partial u_j}{\partial x_1 }\mathbf{e}_1+ \ldots + \frac{\partial u_j}{\partial x_n }\mathbf{e}_n ) = [/tex]

[tex]=n\frac{\partial u_j}{\partial u_i} = [/tex]

[tex]=n\delta^i_j[/tex]

Why am I getting that wrong multiplication byn?

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(Funny) EDIT: since there is a well-known proof in every book which correctly shows that the aforementioned inner-product equals [tex]\delta^i_j[/tex], if no-one manages to find my mistake, our world will have an "amazing" proof that 1=2=...=n for any integer n :)

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# Curvilinear coordinates question

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