Hello,(adsbygoogle = window.adsbygoogle || []).push({});

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?

------------------------------------------------------

(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 :)

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Curvilinear coordinates question

**Physics Forums | Science Articles, Homework Help, Discussion**