ABSTRACT
Let Γ be a simple closed curve in space whose projection in the x, y-plane is the simple closed curve Γ 0 characterized by f 0(x, y) = 0. To simplify our discussion, we take Γ to be entirely above the x, y-plane and described by a well-defined function Z(x, y) for all (x, y) on Γ 0. Let z(x, y) be a surface in space with Γ as its boundary (see Fig. 15.1). Evidently, there are many such surface even if we limit ourselves to only smooth (simply-connected) surface whose projection on the x, y-plane is the entire interior of Γ 0. The surface area for each such surface is given by the double integral () J [ z ] = ∬ A 1 + ( z , x ) 2 + ( z , y ) 2 d x d y , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749821/5a9bd1b0-aa97-4dfe-adb3-8f69a65a6308/content/eq3923.tif"/> https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749821/5a9bd1b0-aa97-4dfe-adb3-8f69a65a6308/content/fig15_1_OB.tif"/>
