(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

[tex]x = \frac{u^{2} + v^{2}}{2}[/tex]

[tex]y = uv[/tex]

[tex]z = z[/tex]

Find the arc length given:

[tex]u(t) = cos(t), v(t) = sin(t), z = \frac{2t^{\frac{3}{2}}}{3}[/tex]

2. Relevant equations

[tex]ds^{2} = dx^{2} + dy^{2} + dz^{2}[/tex]

In curvilinear coordinates thhis becomes

[tex]ds = \sqrt{h^{2}_{1}du^{2}_{1} + h^{2}_{2}du^{2}_{2} + h^{2}_{3}du^{2}_{3}}[/tex]

3. The attempt at a solution

First I need to get the scale factors, so I took the derivative of each x, y, z component.

I came up with:

[tex]dx = udu - vdv[/tex]

[tex]dy = vdu + udv[/tex]

[tex]dz = dz[/tex]

I then found the scale factors,

[tex]h_{1} = h_{u} = \sqrt{u^{2} + v^{2}}[/tex]

[tex]h_{2} = h_{v} = \sqrt{u^{2} + v^{2}}[/tex]

[tex]h_{3} = h_{z} = 1[/tex]

Then we inject the scale factors into the element arc length formula.

[tex]ds = \sqrt{h^{2}_{1}du^{2}_{1} + h^{2}_{2}du^{2}_{2} + h^{2}_{3}du^{2}_{3}}[/tex]

I'm not sure what to do about du1, du2, and du3. Are they just dx, dy and dz? And if so, would this mean I have to integrate 3 times to get the arc length?

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Element of Arc Length Problem

**Physics Forums | Science Articles, Homework Help, Discussion**