# How to find the unit normal vector for typical solid region?

For instance, normal for sphere is easily written as

$$\frac{\langle x,y,z\rangle}{\sqrt{x^2+y^2+z^2}},$$

because the position vector $\langle x,y,z\rangle$ is always perpendicular to tangent plane, and we use $$\sqrt{x^2+y^2+z^2}$$ to make it unit.

But how about other typical solid regions like paraboloid, rectangular box, cylinder, etc.. Is there a general formula?

• Just use differentiation – DHMO Apr 17 '17 at 3:22
• Before you ask for a single general formula for the normal of various regions, can you say whether you know a single general formula for the regions themselves? – Rahul Apr 17 '17 at 3:30

Describe the surface of your region in the form $f(x,y,z)=0$ Then the unit normal vector is given by
$$\hat n= \frac{\vec \nabla f}{ |\vec \nabla f| } = \frac{ \left< \frac { \partial f}{\partial x}, \frac { \partial f}{\partial y}, \frac { \partial f}{\partial z} \right >}{ \sqrt{ ( \frac { \partial f}{\partial x} )^2+( \frac { \partial f}{\partial y} )^2 +( \frac { \partial f}{\partial z})^2 } }$$