The length element in cylindrical coordinates

[stripes]

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}

Homework Equations

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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 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}

Homework Equations

--

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
<br /> dx = \frac{\partial x}{\partial \rho}d\rho + \frac{\partial x}{\partial \theta}d\theta<br />
Squaring both sides gives dx^2.

 

[stripes]

yeah i got it. thanks.