implicit_diff_tangent_plane - Note – Implicit...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

Page1 / 2

implicit_diff_tangent_plane - Note – Implicit...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online