ManDay
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So I wonder why the gradient in coordniates other than cartesian ones bears coefficients. Let's take spherical coordinates for example. We have
(Source) - Sorry if image doesn't work - too lazy to get the TeX right.[/size]
From what I know, I don't see anything that raises cartesian coordinates above any other C/S' (apart from marginal properties such as whether the mapping is reversable). I hence would expect the gradient in spherical coordinates to be
\vec\nabla = \vec{e}_r\frac{d}{dr} + \vec{e}_\phi\frac{d}{d\phi} + \vec{e}_\theta\frac{d}{d\theta},
where \vec{e}_\dots is the unit vector in \vec{g}_{\dots} = \frac{\partial \vec{r}}{\partial \dots} direction (where |g| is not normalized).
After all, the gradient in coordinate q descibes the change of intensity I in q direction. And when I change \vec{r} by a little d\theta, the change in intensity is naturally \frac{dI}{d\phi} and not \frac{1}{r\sin(\phi)}\frac{\partial I}{\partial \theta} (why would it be partial anyway?).
So why is transformed from cartesian coordinates? What validates cartesian coordinates as the origin of all transforms, so to say? I could understand the application of the Leibniz'en chain rule if \nabla to a scalar potential in general coordinates were sought with respects to cartesian coordinates. But we are looking for the gradient in terms of spherical coordinates - so what do cartesian coordinates want to tamper with?best regards
(Source) - Sorry if image doesn't work - too lazy to get the TeX right.[/size]
From what I know, I don't see anything that raises cartesian coordinates above any other C/S' (apart from marginal properties such as whether the mapping is reversable). I hence would expect the gradient in spherical coordinates to be
\vec\nabla = \vec{e}_r\frac{d}{dr} + \vec{e}_\phi\frac{d}{d\phi} + \vec{e}_\theta\frac{d}{d\theta},
where \vec{e}_\dots is the unit vector in \vec{g}_{\dots} = \frac{\partial \vec{r}}{\partial \dots} direction (where |g| is not normalized).
After all, the gradient in coordinate q descibes the change of intensity I in q direction. And when I change \vec{r} by a little d\theta, the change in intensity is naturally \frac{dI}{d\phi} and not \frac{1}{r\sin(\phi)}\frac{\partial I}{\partial \theta} (why would it be partial anyway?).
So why is transformed from cartesian coordinates? What validates cartesian coordinates as the origin of all transforms, so to say? I could understand the application of the Leibniz'en chain rule if \nabla to a scalar potential in general coordinates were sought with respects to cartesian coordinates. But we are looking for the gradient in terms of spherical coordinates - so what do cartesian coordinates want to tamper with?best regards