Question

Find the unit normal to the surface 𝑦 = 𝑥 + 𝑧 3 at the point (1,2,1). 

Answer 1

$$F(x, y, z)=x-y+z^3=0$$

$$F_x=1, F_y=-1, F_z=3z^2$$

Point $(1, 2, 1)$

$$F_x=1, F_y=-1, F_z=3(1)^2=3$$

$$\nabla F=i-j+3k$$

$$|\nabla F|=\sqrt{(1)^2+(-1)^2+(3)^2}=\sqrt{11}$$

$$n=\dfrac{\nabla F}{|\nabla F|}=\dfrac{1}{\sqrt{11}}i-\dfrac{1}{\sqrt{11}}j+\dfrac{3}{\sqrt{11}}k$$