Intuitive / self-apparent derivation of gradient in curvilinear coords

[raxAdaam]

Hi there -

I'm looking for a clear and intuitive explanation of how one obtains the gradient in polar / cylindrical / curvilinear coords.

I do a lot of tutoring, but am finding that the method I've been using (basically chain rule + nature of directional derivative) just doesn't roll with students: the directional derivative approach (i.e. defining the unit tangent vector to the curve) seems really unintuitive to students - either because they've not really seen it or because they've not internalized it.

I don't need a rigorous method that applies to any imaginable coord. system, just something to help them a) remember b) derive and c) understand the gradient (and eventually curl, div etc.) in different coordinate systems. I have some vague recollection of seeing something pretty intuitive a long time ago, but it has slipped my memory and I just use the above described method personally. Everything I've found online so far has been a little too vague about details and unclear with its notation, so now I'm here!

Thanks in advance for the help!



Rax

 

[fresh_42]

I would stress the difference between what is happening and its description via coordinates. The choice of the coordinate system is only a choice of description, not a choice which affects the gradient. Try to make an image.