- #1
wifi
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Problem:
Starting from the gradient of a scalar function T(x,y,z) in cartesian coordinates find the formula for the gradient of T(s,ϕ,z) in cylindrical coordinates.
Solution (so far):
I know that the gradient is given by [itex]\nabla T = \frac{\partial T}{\partial x}\hat{x}+\frac{\partial T}{\partial y}\hat{y}+\frac{\partial T}{\partial z}\hat{z}[/itex]. We must use the chain rule, so we have [itex]\frac{\partial T}{\partial x}(\frac{\partial z}{\partial x})=\frac{\partial T}{\partial s}(\frac{\partial s}{\partial x})+\frac{\partial T}{\partial ϕ}(\frac{\partial ϕ}{\partial x})+\frac{\partial T}{\partial z}(\frac{\partial z}{\partial x})[/itex], [itex]\frac{\partial T}{\partial y}=\frac{\partial T}{\partial s}(\frac{\partial s}{\partial y})+\frac{\partial T}{\partial ϕ}(\frac{\partial ϕ}{\partial y})+\frac{\partial T}{\partial z}(\frac{\partial z}{\partial y})[/itex], and [itex]\frac{\partial T}{\partial x}=\frac{\partial T}{\partial s}(\frac{\partial s}{\partial z})+\frac{\partial T}{\partial ϕ}(\frac{\partial ϕ}{\partial z})+\frac{\partial T}{\partial z}(\frac{\partial z}{\partial z})[/itex].
So then I went and calculated ##\hat{x}## and ##\hat{y}## in terms of ##\hat{s}## and ##\hat{ϕ}##. I got ##\hat{x}=cosϕ\hat{s}-sinϕ\hat{ϕ}## and ##\hat{y}=sinϕ\hat{s}+cosϕ\hat{ϕ}##. Note that ##\hat{x} \cdot \hat{x} =1##, ##\hat{y} \cdot \hat{y}=1##, and ##\hat{x} \cdot \hat{y} =0## - as they should.
Since I need partials of s and ϕ with respect to x and y, I found the inversion formulas ##s=\sqrt{x^2+y^2}## and ##ϕ=sin^{1}(\frac{y}{\sqrt{x^2+y^2}})##. (I used x=s cosϕ and y=s sinϕ)
Obviously ##\frac{\partial ϕ}{\partial z}=0## and ##\frac{\partial s}{\partial z} =0##, but
I'm concerned with how complicated ##\frac{\partial ϕ}{\partial x}## and ##\frac{\partial ϕ}{\partial y}## are going to turn out.
Am I on the right track? Thanks in advance.
Starting from the gradient of a scalar function T(x,y,z) in cartesian coordinates find the formula for the gradient of T(s,ϕ,z) in cylindrical coordinates.
Solution (so far):
I know that the gradient is given by [itex]\nabla T = \frac{\partial T}{\partial x}\hat{x}+\frac{\partial T}{\partial y}\hat{y}+\frac{\partial T}{\partial z}\hat{z}[/itex]. We must use the chain rule, so we have [itex]\frac{\partial T}{\partial x}(\frac{\partial z}{\partial x})=\frac{\partial T}{\partial s}(\frac{\partial s}{\partial x})+\frac{\partial T}{\partial ϕ}(\frac{\partial ϕ}{\partial x})+\frac{\partial T}{\partial z}(\frac{\partial z}{\partial x})[/itex], [itex]\frac{\partial T}{\partial y}=\frac{\partial T}{\partial s}(\frac{\partial s}{\partial y})+\frac{\partial T}{\partial ϕ}(\frac{\partial ϕ}{\partial y})+\frac{\partial T}{\partial z}(\frac{\partial z}{\partial y})[/itex], and [itex]\frac{\partial T}{\partial x}=\frac{\partial T}{\partial s}(\frac{\partial s}{\partial z})+\frac{\partial T}{\partial ϕ}(\frac{\partial ϕ}{\partial z})+\frac{\partial T}{\partial z}(\frac{\partial z}{\partial z})[/itex].
So then I went and calculated ##\hat{x}## and ##\hat{y}## in terms of ##\hat{s}## and ##\hat{ϕ}##. I got ##\hat{x}=cosϕ\hat{s}-sinϕ\hat{ϕ}## and ##\hat{y}=sinϕ\hat{s}+cosϕ\hat{ϕ}##. Note that ##\hat{x} \cdot \hat{x} =1##, ##\hat{y} \cdot \hat{y}=1##, and ##\hat{x} \cdot \hat{y} =0## - as they should.
Since I need partials of s and ϕ with respect to x and y, I found the inversion formulas ##s=\sqrt{x^2+y^2}## and ##ϕ=sin^{1}(\frac{y}{\sqrt{x^2+y^2}})##. (I used x=s cosϕ and y=s sinϕ)
Obviously ##\frac{\partial ϕ}{\partial z}=0## and ##\frac{\partial s}{\partial z} =0##, but
I'm concerned with how complicated ##\frac{\partial ϕ}{\partial x}## and ##\frac{\partial ϕ}{\partial y}## are going to turn out.
Am I on the right track? Thanks in advance.