Chapter 12: Q. 72 (page 918)

Let $w = f (x, y, z)$ be a function of three variables. Prove that when $c_{1} \neq c_{2}$ , the level surfaces defined by the equations $f (x, y, z) = c_{1}$ and $f (x, y, z) = c_{2}$ do not intersect

Short Answer

Expert verified

We proved by contradictions that the equations do not intersect

Step by step solution

Given information

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

Explanation

Let the function can be given as

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

Let these two function be intersecting and point of intersection is $(x_{0}, y_{0}, z_{0})$

Substituting the equations 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 is equal then the right hand side should also be equal and we get,

$c_{1} = c_{2}$

But we are given $c_{1} \neq c_{2}$

hence our assumption is wrong

The equations do not intersect

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Let $w = f (x, y, z)$ be a function of three variables. Prove that when $c_{1} \neq c_{2}$ , the level surfaces defined by the equations $f (x, y, z) = c_{1}$ and $f (x, y, z) = c_{2}$ do not intersect

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 when c1≠c2, the level surfaces defined by the equations f(x,y,z)=c1 andf(x,y,z)=c2 do not intersect

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

Applied Mathematics

Pure Maths

Mechanics Maths

Theoretical and Mathematical Physics

Probability and Statistics

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Study anywhere. Anytime. Across all devices.

Let $w = f (x, y, z)$ be a function of three variables. Prove that when $c_{1} \neq c_{2}$ , the level surfaces defined by the equations $f (x, y, z) = c_{1}$ and $f (x, y, z) = c_{2}$ do not intersect