# The length element in cylindrical coordinates

## Homework Statement

Show that in cylindrical coordinates

$x = \rho cos \theta$
$y = \rho sin \theta$
$z = z$

the length element ds is given by

$ds^{2} = dx^{2} + dy^{2} + dz^{2} = d \rho^{2} + \rho^{2} d \theta ^{2} + dz^{2}$

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## The Attempt at a Solution

Notice $\rho = \sqrt{x^{2} + y^{2}}$

I have found expressions for ∂x/∂θ, ∂x/∂ρ, ∂y/∂θ, ∂y/∂ρ, ∂ρ/∂x, ∂ρ/∂y, ∂θ/∂x, ∂θ/∂y, and of course it is trivial that dz = dz. Given all of these, how do i start using chain rule and total derivatives to find ds2? I guess because i have 8 partials but end up only ρ, θ, and z, i don't really know which ones to start with.

Any help is greatly appreciated.

pasmith
Homework Helper

## Homework Statement

Show that in cylindrical coordinates

$x = \rho cos \theta$
$y = \rho sin \theta$
$z = z$

the length element ds is given by

$ds^{2} = dx^{2} + dy^{2} + dz^{2} = d \rho^{2} + \rho^{2} d \theta ^{2} + dz^{2}$

--

## The Attempt at a Solution

Notice $\rho = \sqrt{x^{2} + y^{2}}$

I have found expressions for ∂x/∂θ, ∂x/∂ρ, ∂y/∂θ, ∂y/∂ρ, ∂ρ/∂x, ∂ρ/∂y, ∂θ/∂x, ∂θ/∂y, and of course it is trivial that dz = dz. Given all of these, how do i start using chain rule and total derivatives to find ds2? I guess because i have 8 partials but end up only ρ, θ, and z, i don't really know which ones to start with.

Any help is greatly appreciated.

Start from
$$dx = \frac{\partial x}{\partial \rho}d\rho + \frac{\partial x}{\partial \theta}d\theta$$
Squaring both sides gives $dx^2$.

yeah i got it. thanks.