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Cylindrical polar coordinates

  • Thread starter chwala
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  • #1
chwala
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Homework Statement


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?[/B]


Homework Equations




The Attempt at a Solution


##h1=|∂f/∂ℝ|= sin θ,
h2=|∂f/∂θ|=ℝ cos θ
h3=|∂f/∂z|=2z##
advice [/B]
 

Answers and Replies

  • #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##.
 
  • #3
chwala
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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:

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