example of integration with respect to surface area on a helicoid

In this example, we shall consider itegration with respect to surface area on the helicoid.

The helicoid may be parameterized as follows:

$x=u\sin v$

$y=u\cos v$

$z=cv$

(The constant $c$ may be thought of as the “pitch of the screw”.) Computing derivatives and appying trigonometric identities, we obtain

$\frac{\partial(x,y)}{\partial(u,v)}=\left|\begin{matrix}\sin v&u\cos v\\ \cos v&-u\sin v\end{matrix}\right|=-u$

$\frac{\partial(y,z)}{\partial(u,v)}=\left|\begin{matrix}\cos v&-u\sin v\\ 0&c\end{matrix}\right|=c\cos v$

$\frac{\partial(z,x)}{\partial(u,v)}=\left|\begin{matrix}0&c\\ \sin v&u\cos v\end{matrix}\right|=-c\sin v.$

From this we have

$\sqrt{\left(\frac{\partial(x,y)}{\partial(u,v)}\right)^{2}+\left(\frac{% \partial(y,z)}{\partial(u,v)}\right)^{2}+\left(\frac{\partial(z,x)}{\partial(u% ,v)}\right)^{2}}=$

$\sqrt{u^{2}+c^{2}\cos^{2}v+c^{2}\sin^{2}v}=\sqrt{u^{2}+c^{2}}$

so we can compute area integrals over helicoids as follows

$\int_{S}f(u,v)\,d^{2}A=\int f(u,v)\sqrt{c^{2}+u^{2}}\>du\,dv$

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Title	example of integration with respect to surface area on a helicoid
Canonical name	ExampleOfIntegrationWithRespectToSurfaceAreaOnAHelicoid
Date of creation	2013-03-22 14:58:01
Last modified on	2013-03-22 14:58:01
Owner	rspuzio (6075)
Last modified by	rspuzio (6075)
Numerical id	7
Author	rspuzio (6075)
Entry type	Example
Classification	msc 28A75