How to Perform Operations on Vector Fields F and G?

[Obstacle1]

Supposing we have as 2 vector fields:

F = x^2i + 2zj +3k
and
G = r^2e_r + 2\cos\Theta e_{\Theta} + 3\sin(2\phi) e_\phi

how do i perform the following operations on them?

- F\cdot r

- F\times r

- |G|

- G\cdot r

 

[robphy]

Can you write \hat r in terms of rectangular components and of spherical-polar coordinates?

 

[Obstacle1]

Can you write \hat r in terms of rectangular components and of spherical-polar coordinates?

No, sorry. How do i do that??

 

[shoehorn]

No, sorry. How do i do that??

Suppose that you are in three dimensions. You can use standard Cartesian coordinates (x,y,z) or spherical polar coordinates (r,\theta,\phi) to describe this three-dimensional space in a convenient manner.

To begin finding a solution to your problem, how are the coordinates (r,\theta,\phi) related to the coordinates (x,y,z)?

Now how are the basis vectors \{\hat{e}_x,\hat{e}_y,\hat{e}_z\} for the Cartesian coordinate system related to the basis vectors \{\hat{e}_r,\hat{e}_\theta,\hat{e}_\phi\} of the spherical polar coordinate system?