Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP
Posting-Version: version B 2.10.2 GARFIELD 20/11/84; site garfield.UUCP
Path: utzoo!utcsri!garfield!robertj
From: robertj@garfield.UUCP
Newsgroups: net.math
Subject: Re: Surface Area Query
Message-ID: <4407@garfield.UUCP>
Date: Wed, 20-Nov-85 08:40:57 EST
Article-I.D.: garfield.4407
Posted: Wed Nov 20 08:40:57 1985
Date-Received: Thu, 21-Nov-85 19:43:00 EST
References: <749@h-sc1.UUCP>
Sender: paulc@garfield.UUCP
Reply-To: robertj@garfield.UUCP (Robert Janes)
Distribution: net
Organization: Memorial U. of Nfld. C.S. Dept., St. John's
Lines: 36
Summary: 

In article <749@h-sc1.UUCP> riggsby@h-sc1.UUCP (andrew riggsby) writes:
>Is there a general method for determining the surface area of an arbitrary sur- face or solid similar to (or not similar, for that matter) using displacement
>to determine the volume of a solid.  Note that this includes 2-d surfaces in
>E^3.
>
>Thanks.
>
>                               Andrew Riggsby
>                              riggsby@harvunxu
>    

If the surface is given in parametric form and is of the class C^2 then
there is a method that I know of which is good for arbitrary dimensions.
It involves the (n-1)th integral (i.e. in E^3 this is the double integral)
of the determinant of the metric tensor of the surface. In more detail,
let the surface be parametrized by:

	R = Xi = Xi(U1,U2,...,U(n-1))

let Ri designate the partial derivative of R with respect to the ith
coordinate,Ui.
Then Gij = Ri.Rj where Ri.Rj is the usual scalar product or dot product.
All such Gij define a matrix with dterminant G. If we let I denote the
the (n-1)th integral (that is double integrals for E^3, triple integrals
for E^4) then if S is the surface area of the surface in question and
is given by:

	S = I{ (G)^(1/2) dU1dU2dU3...dU(n-1) } 

over the region in question (that is the set on which each Ui is defined ).
This I think is what you are looking for and I think that it is correct 
inwhat it is saying. If not humblest apologies. Reference: Manfredo P.
do Carmo, The Differential Geometry of Curves and Surfaces.

					Cheers
					Robert Janes