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Chapter 14: Q. 3 (page 1140)

Write two different normal vectors for a smooth surface S given by (x, y, g(x, y)) at the point(x0,y0,g(x0,y0)).

Short Answer

Expert verified

TherequiredtwodifferentnormalvectorsforasmoothsurfaceSdescribedby,(x,y,g(x,y))atthepoint(x0,y0,g(x0,y0))are-∂g∂x,-∂g∂y,1and∂g∂x,∂g∂y,-1

Step by step solution

01

Step 1. Given 

smooth surface S given by (x, y, g(x, y)) at the point(x0,y0,g(x0,y0)).

02

Step 2. Objective

TheobjectiveistocalculatethetwodifferentnormalvectorsforasmoothsurfaceSdescribedby(x,y,g(x,y))atthepoint(x0,y0,g(x0,y0)).Aparametrizedsurfaceof(x,y,g(x,y))intermsofxandyis,r(x,y)=x,y,g(x,y)).Then,thedesirednormalvectorwillbe,n=±(rx+ry)

03

Step 3. Calculation

NowfindrxandryByabovestep,thesurfaceSisparametrizedbyasfollows,r(x,y)=x,y,g(x,y)).then,rx=∂∂xr(x,y)=∂∂xx,y,g(x,y))=1,0,∂g∂x.and,ry=∂∂yr(x,y)=∂∂yx,y,g(x,y))=1,0,∂g∂y.Nextfindrx×ry.rx×ry=1,0,∂g∂x×1,0,∂g∂y.=ijk10∂g∂x01∂g∂y=0.∂g∂y-∂g∂x.1i-1.∂g∂y-∂g∂x.0j+1.1-0.0k=-∂g∂xi-∂g∂yj+k=-∂g∂x,-∂g∂y,1.

04

Step 4. Result

Thenthenormalvectorwillbe,n=±(rx+ry),orn=±-∂g∂x,-∂g∂y,1impliesthat,n=-∂g∂x,-∂g∂y,1andn=--∂g∂x,-∂g∂y,1impliesthat,n=-∂g∂x,-∂g∂y,1andn=∂g∂x,∂g∂y,-1ThereforetherequiredtwodifferentnormalvectorsforasmoothsurfaceSdescribedby,(x,y,g(x,y))atthepoint(x0,y0,g(x0,y0))are-∂g∂x,-∂g∂y,1and∂g∂x,∂g∂y,-1

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