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 $f (x, y, z) = c_{1}$ and $f (x, y, z) = c_{2}$ intersect, then the surfaces are identical

Short Answer

Expert verified

We proved that the equations are identical

Step by step solution

Given information

We are given a function of three variables $w = f (x, y, z)$

Explanation

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

$a x + b y + d z = c_{1} a n d a x + b y + d z = c_{2}$

Let the planes intersect and point of intersection be $(x_{0}, y_{0}, z_{0})$

Hence the equation becomes

$a x_{0} + b y_{0} + d z_{0} = c_{1} a n d a x_{0} + b y_{0} + d z_{0} = c_{2}$

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

We are also given that $c_{1} = c_{2}$

Hence the equations are identical

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Let w = f(x, y, z) be a function of three variables. Prove that if the level surfaces defined by the equations $f (x, y, z) = c_{1}$ and $f (x, y, z) = c_{2}$ intersect, then the surfaces are identical

Short Answer

Step by step solution

Given information

Explanation

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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

Step by step solution

Given information

Explanation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Mechanics Maths

Pure Maths

Statistics

Geometry

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

Let w = f(x, y, z) be a function of three variables. Prove that if the level surfaces defined by the equations $f (x, y, z) = c_{1}$ and $f (x, y, z) = c_{2}$ intersect, then the surfaces are identical