Understanding Scale Factors in Cylindrical Polar Coordinates

[chwala]

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]

 

[Orodruin]

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 $\rho$.

 

[chwala]

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!