Let [itex] \vec{r} = \vec{r}(q_1,\ldots,q_n) [/itex].(adsbygoogle = window.adsbygoogle || []).push({});

Is the following ALWAYS true?

[tex]

\frac{\partial \vec{r}}{\partial q_i} \cdot \frac{\partial \vec{r}}{\partial q_j} = \delta_{ij}

[/tex]

Edit: Perhaps I should ask if it is zero when [itex] i \neq j [/itex] rather than saying that it is [itex] 1 [/itex] when [itex] i = j [/itex]

I guess I am wondering if this statement is true ONLY IF we have an orthonormal coordinate system. Does this hold for nonorthonormal coordinate systems? Does it hold if our vectors are not normalized (I would think not)?

Also, when we have a position vector....do we always treat it in rectangular coordinates and then express each of the coordinates in terms of our generalized coordinates. Let me clarify....for 3D, we have [itex] \vec{r} = (x,y,z) [/itex]. Do we always express x, y, and z in terms of our generalized coordinates (for example [itex] x = r \cos \theta \sin \phi [/itex])? It seems like we do. I am wondering if we ever think of [itex] \vec{r} [/itex] strictly in terms of the q---so, [itex] \vec{r} = (q_1,\ldots,q_N) [/itex]. I believe that the answer is no...since [itex] \vec{r} [/itex] must have proper units...and our generalized coordinates might not have this "feature".

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# Partial Derivatives of Position Vector

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