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Chapter 12: Q. 73 (page 918)

Let w = f(x, y, z) be a function of three variables. Prove that if the level surfaces defined by the equations f(x,y,z)=c1and f(x,y,z)=c2 intersect, then the surfaces are identical

Short Answer

Expert verified

We proved that the equations are identical

Step by step solution

01

Given information

We are given a function of three variablesw=f(x,y,z)

02

Explanation

We are given a function of three variables which we can write as

ax+by+dz=c1andax+by+dz=c2

Let the planes intersect and point of intersection be (x0,y0,z0)

Hence the equation becomes

ax0+by0+dz0=c1andax0+by0+dz0=c2

As the left hand side of the equation is equal right hand side should also be

We are also given that c1=c2

Hence the equations are identical

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