Q. 72 - Calculus

Question

Let $w=f(x, y,z)$ be a function of three variables. Prove that when $c_1\ne c_2$, the level surfaces defined by the equations $f(x,y,z)=c_1$ and $f(x,y,z)=c_2$ do not intersect 

Step-by-Step Solution

Answer

We proved by contradictions that the equations do not intersect

Step 1: Given information

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

Step 2: Explanation

Let the function can be given as

$ax+by+dz=c_1 and ax+by+dz=c_2$

Let these two function be intersecting and point of intersection is $(x_0,y_0,z_0)$

Substituting the equations becomes

$a{x}_0+b{y}_0+d{z}_0=c_1 and a{x}_0+b{y}_0+d{z}_0=c_2$

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 $c_1\ne c_2$

hence our assumption is wrong

The equations do not intersect