Vector Operations in Polar Coordinates?

[Poweranimals]

Do you think you could do vector operations in polar coordinates?

 

[Galileo]

When you're in \mathbb{R}^3 and you want to designate the tip of a vector by giving three coordinates, then you can use spherical coordinates (or polar coordinates in \mathbb{R}^2) or any other coordinate system.

But using rectangular coordinates is much more convenient, because the vector notation in component form will mean the first component times the first basis-vector, the second component times the second basis-vector and so on. This can't be done in polar/spherical coordinates.
Also, adding vectors component-wise, and things like the dot-product are of no use either.

So much of the reason why you use vectors will be lost when going to spherical/polar coordinates.

In short, with vectors: use a linear coordinate system.

 

[James R]

Depends what you're doing. In physics, it is often convenient to use spherical polar coordinates for vector fields, particular if the field is spherically symmetrical. If you have cylindrical symmetry, cylindrical polar coordinates are often useful.

You can certainly write grad f, div V, and curl V in terms of their polar components.