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Cylindrical polar coordinates

  1. Apr 22, 2016 #1
    1. The problem statement, all variables and given/known data
    Using the cylindrical polar co ordinates ##(ℝ,θ,z)## calculate the gradient of ##f=ℝ sin θ + z^2##

    the textbook says that the scale factors are ## h1=1, h2=ℝ & h3=1##

    how did they arrive at this?



    2. Relevant equations


    3. The attempt at a solution
    ##h1=|∂f/∂ℝ|= sin θ,
    h2=|∂f/∂θ|=ℝ cos θ
    h3=|∂f/∂z|=2z##
    advice
     
  2. jcsd
  3. Apr 22, 2016 #2

    Orodruin

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    Your textbook should then also mention the definition of the scale factors and how the gradient operator is expressed in curvilinear coordinate systems?

    Also, do not use ##\mathbb R## to denote anything other than the real numbers, the default notation for the radial coordinate in polar or cylinder coordinates would generally be ##r## or ##\rho##.
     
  4. Apr 23, 2016 #3
    Thanks a lot the scale factor is defined as follows,....a small preview
    the cartesian system and curvilinear system are 1:1
    where## x(u1,u2,u3)= u(x1,x2,x3)##
    where x defines cartesian and u defines curvilinear coordinate system.
    It follows that
    ## x= cos θ , y=sin θ , z=z##.
    In the conversion from cartesian system ##(x,y,z)## to curvilinear system ##(r,θ,z)##
    the displacement ## r= xi+yj+zk##
    small displacement ## dr= (∂f/∂u1)dr1 + (∂f/∂u2)dr2+ (∂f/∂u3)dr3## i will check this.
    ok i will restrict myself to the equations without going into a lot of details,
    we have ## f(R,θ,z)= R cos θi+ Rsin θj + zk##
    The scale facors are given as follows
    ## h1= mod ∂f/dR=1 & h2= mod ∂f/∂θ= R &
    h3= mod ∂f/∂z = 1 ##

    thanks greetings from Africa chikhabi!
     
    Last edited: Apr 23, 2016
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