Basis vectors in different coordinate systems

Question

I was wondering about the following: The basis vectors used for the gradient, and curl in cylindrical and spherical coordinates are defined to be with unit vectors, why is that so? What if the basis vectors weren't made into unit length, what would be the issue?

Answer 1

If I understand you right, there isn't really an issue with using other vectors, but the point of making them unit length is convenience. For example, in $\mathbb{R}^3 $, you could use the basis $ \{ (0, 0, 2), (0, 2, 0), (2, 0, 0) \} $ and you would be perfectly fine in doing so, but then you'd be multiplying all the coordinates by $1/2 $. It is more convenient if you don't have to do this, so you normalize to unit length. This convenience widely applies not just to your examples or my example--same with the definition of wedge products.