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

  1. Sep 11, 2014 #1
    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, [tex]\vec{F}=F\hat{z}[/tex], where [tex]\hat{z}[/tex] is the unit vector in the z direction. Express [tex]\vec{F}[/tex] 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 [tex]\hat{z}[/tex] to spherical coordinates.

    I found the answer to be [tex]\hat{z} = cos(\phi)\hat{r}-sin(\phi)\hat{\phi}[/tex] 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 [tex]\vec{F}=F\hat{z} = F(cos(\phi)\hat{r}-sin(\phi)\hat{\phi})[/tex]

    thanks!
     
  2. jcsd
  3. Sep 11, 2014 #2
    Yes, you are done! resolving the Cartesian unite vector z into its spherical polar coordinates yields
    [tex]\hat{z} = cos(\phi)\hat{r}-sin(\phi)\hat{\phi}[/tex]

    you just need to substitute the above equation into the relation [tex]\vec{F}=F\hat{z}[/tex].
     
  4. Sep 11, 2014 #3
    So: [tex]\vec{F}=F\hat{z} = F(cos(\phi)\hat{r}-sin(\phi)\hat{\phi})[/tex] is the answer?

    But my question was if someone can explain why [tex]\hat{z}= cos(\phi)\hat{r}-sin(\phi)\hat{\phi}[/tex]
     
  5. Sep 11, 2014 #4
    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.
     
  6. Sep 11, 2014 #5
    physics student
     
  7. Sep 11, 2014 #6

    ehild

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    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
     

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  8. Sep 11, 2014 #7

    BvU

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    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.
     
  9. Sep 11, 2014 #8

    ehild

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    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
     
  10. Sep 11, 2014 #9

    BvU

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    Fully agree. What triggered me is the
     
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