Chapter 4: Partial Differentiation
Section 4.6: Surface Normal and Tangent Plane
Table 4.6.1 lists expressions for a vector N, normal to a surface defined either explicitly by and equation of the form z=fx,y; or implicitly, by an equation of the form fx,y,z=c, where c is a real constant.
Surface defined explicitly
Surface defined implicitly
Table 4.6.1 Surface normals for surfaces defined explicitly and implicitly
If the gradient ∇f is normal to the level surface defined implicitly by an equation of the form fx,y,z=c, then writing the equation z=fx,y for the explicitly given surface in the implicit form z−fx,y=0, leads to the surface normal
At P:2,−3 on the surface defined by z=fx,y≡5−x2/3−y2/2, obtain and draw both the normal and tangent plane.
At P:a,b on the surface defined by z=fx,y, obtain an equation for the tangent plane in the form z=….
Derive the form of N for the surface given explicitly by z=fx,y. (See Table 4.6.1.)
At P:1,1,1 on the surface defined by fx,y,z≡x2+2 y2+3 z2=6, obtain and draw both the normal and tangent plane.
At P:a,b,c on the surface defined implicitly by fx,y,z=0, obtain an equation for the tangent plane in the form z=….
Derive the form of N for the surface given implicitly by fx,y,z=c, where c is a real constant. (See Table 4.6.1.)
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