Spherical coordinates?

1. Jun 2, 2006

pivoxa15

What is equivalent to the unit k (vector in cartesian coords) in spherical coordinates? And why?

z=rcos(t)

Last edited: Jun 2, 2006
2. Jun 2, 2006

MalayInd

if the vector is in three dimensions, one more variable(of spherical) is required to define j.

3. Jun 2, 2006

dav2008

Last edited: Jun 2, 2006
4. Jun 2, 2006

pivoxa15

I made a mistake which has been corrected, it should be the unit k vector.

In cartesian, it is (0,0,1). What is it in spherical (0,0,what)?

5. Jun 3, 2006

arildno

For the most common choice of spherical polar coordinates,
$$x=r\sin\phi\cos\theta,y=r\sin\phi\sin\thea,z=r\cos\phi,0\leq{r},0\leq\phi\leq\pi,0\leq\theta\leq{2}\pi[/itex] we have the following unit vetors relations: [tex]\vec{i}_{r}=\sin\phi(\cos\theta\vec{i}+\sin\theta\vec{j})+\cos\phi\vec{k}$$
$$\vec{i}_{\phi}=\frac{\partial}{\partial\phi}\vec{i}_{r}=\cos\phi(\cos\theta\vec{i}+\sin\theta\vec{j})-\sin\phi\vec{k}$$
$$\vec{i}_{\theta}=\frac{1}{\sin\phi}\frac{\partial}{\partial\theta}\vec{i}_{r}=-\sin\theta\vec{i}+\cos\theta\vec{j}$$
Solving for the Cartesian unit vectors we gain, in particular:
$$\vec{k}=\cos\phi\vec{i}_{r}-\sin\phi\vec{i}_{\phi}$$
That is of course equal to the coordinate transformation:
$$(0,0,1)\to(\cos\phi,0,\sin\phi)$$
In order to find the correct expressions for the other two unit Cartesian vectors, utilize the intermediate result:
[tex]\sin\phi\vec{i}_{r}+\cos\phi\vec{i}_{\phi}=\vec{i}_{\hat{r}}=\cos\theta\vec{i}+\sin\theta\vec{j}[/itex]
$\vec{i}_{\hat{r}},\vec{i}_{\theta}$ are polar coordinate vectors in the horizontal plane.

Last edited: Jun 3, 2006