# Spherical coordinates?

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

z=rcos(t)

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## Answers and Replies

pivoxa15 said:
What is equivalent to the unit j (vector in cartesian coords) in spherical coordinates? And why?

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

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)?

arildno
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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.

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