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

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

Figure 17

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.