(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?

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# Homework Help: Element of Arc Length Problem

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