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Unformatted text preview: v u r r is a vector normal to the surface. If v u r r for any (u,v) in D, the surface is called smooth and will have a tangent plane. In other words, the surface will have no sharp points or cusps. Area of a Parametric Surface Let S be a smooth parametric surface r (u,v) = x(u,v) I +y(u,v) j + z(u,v) k defined over an open region D in the uv-plane. If each point on the surface S corresponds to exactly one point in the domain D, the surface area of S is given by Surface area = = D v u S dA r r dS Where k u z j u y i u x r u + + = and k v z j v y i v x r v + + = . Example: Find the area of the part of the paraboloid k u j v u i v u v u r 2 sin 2 cos 2 ) , ( + + = where 2 u and 2 v ....
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