Consider a vector A = Arc rc + AΦ Φ + Az z at a point (rc, Φ, z) in cylindrical coordinates. Determine its location and representation in Cartesian coordinates.

Question

Consider a vector A = [latex]A_{r_{c}} \hat{r}_{c} + A_{Φ} \hat{Φ} + A_{z} \hat{z}[/latex] at a point ([latex]r_{c}[/latex], Φ, z) in cylindrical coordinates. Determine its location and representation in Cartesian coordinates.

Accepted Answer

From Table A.1 the location in Cartesian coordinates is x = [latex]r_{c}[/latex] cos(Φ), y = [latex]r_{c}[/latex] sin(Φ), z = z. Similarly, the vector representation is

Table A.1 Coordinate transformations for Cartesian coordinates

Spherical	Cylindrical	Cartesian
= [latex]sin(θ) cos(Φ)\hat{r}_{s} +cos(θ)cos(Φ)\hat{θ} - sin(Φ)\hat{Φ}[/latex]	= [latex]cos(Φ)\hat{r}_{c} - sin(Φ)\hat{Φ}[/latex]	[latex]\hat{x}[/latex]
= [latex]sin(θ) sin(Φ)\hat{r}_{s} +cos(θ)sin(Φ)\hat{θ} + cos(Φ)\hat{Φ}[/latex]	= [latex]sin(Φ)\hat{r}_{c} + cos(Φ)\hat{Φ}[/latex]	[latex]\hat{y}[/latex]
= [latex]cos(θ)\hat{r} -sin(θ)\hat{θ}[/latex]	= [latex]\hat{z}[/latex]	[latex]\hat{z}[/latex]
= [latex]r_{s}[/latex] sin(θ)cos(Φ)	= [latex]r_{c}[/latex]cos(Φ)	x
= [latex]r_{s}[/latex] sin(θ)sin(Φ)	= [latex]r_{c}[/latex]sin(Φ)	y
= [latex]r_{s}[/latex]cos(θ)	= z	z
= [latex]A_{r_{s}}sin(θ)cos(Φ) +A_{θ}cos(θ)cos(Φ) -A_{Φ}sin(Φ)[/latex]	= [latex]A_{r_{c}}cos(Φ) -A_{Φ}sin(Φ)[/latex]	[latex]A_{x}[/latex]
= [latex]A_{r_{s}}sin(θ)sin(Φ) +A_{θ}cos(θ)sin(Φ) + A_{Φ}cos(Φ)[/latex]	= [latex]A_{r_{c}}sin(Φ) -A_{Φ}cos(Φ)[/latex]	[latex]A_{y}[/latex]
= [latex]A_{r_{s}}cos(θ) - A_{θ}sin(θ)[/latex]	= [latex]A_{z}[/latex]	[latex]A_{z}[/latex]

A = [latex]\underbrace{(A_{r_{c}}cos(Φ) -A_{Φ}sin(Φ))}_{A_{x}} \hat{x} + \underbrace{(A_{r_{c}}sin(Φ) -A_{Φ}cos(Φ))}_{A_{y}} \hat{y} + A_{z}\hat{z}.[/latex]

Spherical	Cylindrical	Cartesian
= $sin(θ) cos(Φ)\hat{r}_{s} +cos(θ)cos(Φ)\hat{θ} – sin(Φ)\hat{Φ}$	= $cos(Φ)\hat{r}_{c} – sin(Φ)\hat{Φ}$	$\hat{x}$
= $sin(θ) sin(Φ)\hat{r}_{s} +cos(θ)sin(Φ)\hat{θ} + cos(Φ)\hat{Φ}$	= $sin(Φ)\hat{r}_{c} + cos(Φ)\hat{Φ}$	$\hat{y}$
= $cos(θ)\hat{r} -sin(θ)\hat{θ}$	= $\hat{z}$	$\hat{z}$
= $r_{s}$ sin(θ)cos(Φ)	= $r_{c}$ cos(Φ)	x
= $r_{s}$ sin(θ)sin(Φ)	= $r_{c}$ sin(Φ)	y
= $r_{s}$ cos(θ)	= z	z
= $A_{r_{s}}sin(θ)cos(Φ) +A_{θ}cos(θ)cos(Φ) -A_{Φ}sin(Φ)$	= $A_{r_{c}}cos(Φ) -A_{Φ}sin(Φ)$	$A_{x}$
= $A_{r_{s}}sin(θ)sin(Φ) +A_{θ}cos(θ)sin(Φ) + A_{Φ}cos(Φ)$	= $A_{r_{c}}sin(Φ) -A_{Φ}cos(Φ)$	$A_{y}$
= $A_{r_{s}}cos(θ) – A_{θ}sin(θ)$	= $A_{z}$	$A_{z}$

Question A.6.1: Consider a vector A = Arc rc + AΦ Φ + Az z at a point (rc, Φ...

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