Describing Level Surfaces

Describe the level surfaces of the function \(w=f(x,y,z)=2x+3y+z\).

Solution  The level surfaces are the graphs of \[ 2x+3y+z=c \]

This is a collection of parallel planes, each plane having the vector \( \mathbf{N}=2\mathbf{i}+3\mathbf{j}+\mathbf{k}\) as normal. See Figure 17.