Q. 73 - Calculus

Question

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 

Step-by-Step Solution

Answer

We proved that the equations are identical

Step 1: Given information

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

Step 2: Explanation

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

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

Let the planes intersect and point of intersection be $(x_0,y_0,z_0)$

Hence the equation becomes

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

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