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Unformatted text preview: Note – Implicit Differentiation [email protected] A surface in R 3 is represented explicitly as z = f ( x,y ) for some function f : R 2 → R . It is not always the case, however, that we can write the surface as z = f ( x,y ). For example, the unit sphere is represented most naturally by the equation x 2 + y 2 + z 2 = 1 and we could then attempt to solve for z as a function of x and y . However, it is clear that there is no single function f ( x,y ) such that z = f ( x,y ) defines a sphere since for each x and y value there are two z values on the sphere. Despite this, we can say that around any point on the sphere, we get a surface z = f ( x,y ), and furthermore the surface is smooth. A surface in R 3 is therefore often represented implicitly – that is there is an equation g ( x,y,z ) = 0 which defines the surface. For the sphere it was g ( x,y,z ) = x 2 + y 2 + z 2 1 = 0. While we might not be able to solve explicitly for z in terms of x and y , it is still reasonable to ask if parts of the surface are smooth and can be solved for...
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 Spring '07
 Enright
 Math, Expression, Continuous function, implicit function theorem

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