0
$\begingroup$

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?

$\endgroup$
  • $\begingroup$ Just use differentiation $\endgroup$ – DHMO Apr 17 '17 at 3:22
  • 1
    $\begingroup$ 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? $\endgroup$ – Rahul Apr 17 '17 at 3:30
0
$\begingroup$

Describe the surface of your region in the form $f(x,y,z)=0$ Then the unit normal vector is given by

$$\hat n= \frac{\vec \nabla f}{ |\vec \nabla f| } = \frac{ \left< \frac { \partial f}{\partial x}, \frac { \partial f}{\partial y}, \frac { \partial f}{\partial z} \right >}{ \sqrt{ ( \frac { \partial f}{\partial x} )^2+( \frac { \partial f}{\partial y} )^2 +( \frac { \partial f}{\partial z})^2 } }$$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.