# A vector F=Fz, where z is unit vector, expressed in sphereical coord.

1. Sep 11, 2014

1. The problem statement, all variables and given/known data

So the full problem reads: A vector F has the same magnitude and direction at all points in space. Choose the z-axis parallel to F. Then , in Cartesian coordinates, $$\vec{F}=F\hat{z}$$, where $$\hat{z}$$ is the unit vector in the z direction. Express $$\vec{F}$$ in spherical coordinates.

2. Relevant equations

I don't know?

3. The attempt at a solution

Well pretty much it wants me to express this vector field that is parallel with the z-axis, which really means converting $$\hat{z}$$ to spherical coordinates.

I found the answer to be $$\hat{z} = cos(\phi)\hat{r}-sin(\phi)\hat{\phi}$$ through an online resource I really didn't understand. No images, nothing. Can someone please explain how you can convert the unit vectors to spherical?

And would the answer be $$\vec{F}=F\hat{z} = F(cos(\phi)\hat{r}-sin(\phi)\hat{\phi})$$

thanks!

2. Sep 11, 2014

### PaulDirac

Yes, you are done! resolving the Cartesian unite vector z into its spherical polar coordinates yields
$$\hat{z} = cos(\phi)\hat{r}-sin(\phi)\hat{\phi}$$

you just need to substitute the above equation into the relation $$\vec{F}=F\hat{z}$$.

3. Sep 11, 2014

So: $$\vec{F}=F\hat{z} = F(cos(\phi)\hat{r}-sin(\phi)\hat{\phi})$$ is the answer?

But my question was if someone can explain why $$\hat{z}= cos(\phi)\hat{r}-sin(\phi)\hat{\phi}$$

4. Sep 11, 2014

### PaulDirac

OK. but what's your background in math? Are you a physics student or ? I can refer you to the book Arfken where you'll find how to change the Cartesian coordinates into spherical polar ones.

5. Sep 11, 2014

physics student

6. Sep 11, 2014

### ehild

See the picture. The unit vectors of the polar coordinates change with the position. At point P, $\vec e_r$ or $\hat r$ is the unit vector along the radial vector pointing to P from the origin. $\vec e_Φ$ or $\hat Φ$ is the unit vector perpendicular to $\hat r$ in the plane of the z axis and OP.

If you have a vector $\vec F$ at P, its projection onto the direction of the unit vectors are its components in polar coordinates.

ehild

#### Attached Files:

• ###### polarcomp.JPG
File size:
13.8 KB
Views:
68
7. Sep 11, 2014

### BvU

Yes, but just to harmonize this with future exploits, be aware that naming of spherical coordinates is almost always
$\theta$ polar angle : $\arccos (\hat z \cdot \hat r)$
$\phi$ azimuthal angle : $\arccos (\hat x \cdot \hat r)$
i.e. just the other way around. Better get used to that.

8. Sep 11, 2014

### ehild

I would not say "almost always".

On the first page of Google results, Wiki and Hyperphysics and http://www.nyu.edu/classes/tuckerman/adv.chem/lectures/math_prelims/node12.html were pro, but the following were contra.

http://mathworld.wolfram.com/SphericalCoordinates.html

http://tutorial.math.lamar.edu/Classes/CalcIII/SphericalCoords.aspx

http://www.geom.uiuc.edu/docs/reference/CRC-formulas/node42.html

The OP should us those notations as he was taught.

ehild

9. Sep 11, 2014

### BvU

Fully agree. What triggered me is the