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?