Unit Normal Vector to a Cylinder

[Mangala]

Homework Statement

I need to find the unit normal vector to x²+z²=16. I can find the unit normal vector to every shape but a cylinder, and I can't seem to see how to do it.

Homework Equations

$$\nabla$$F=F_xi+F_yj+F_zk

The Attempt at a Solution

I know that I need to find the gradient of the surface equation. I've considered that this gradient may be i-2x-2z. Or am I completely wrong?

If you need more information please say what. Thank you!

 

[Dick]

You aren't really using your own "Relevant Equation". The level surface equation for your cylinder is F(x,y,z)=x^2+z^2-16=0. Now what is grad(F) again?

 

[n1person]

So if the equation of the cylinder is x^2+z^2=16, i would f(x,y,z)=x^2+z^2, and find the gradient from there.

Then you also have the constraint that it has to be on the cylinder x^2+z^2=16. So I would parametrize that surface, and then plug that parametrization into the gradient to get the equation for the vectors. Then you need to divide by the norm in order to get unit vectors, (which i think might end up very nicely!).

Alternatively, the way i would do it, is just parametrize the cylinder off the bat, then take the derivative, then normalize it (but one doesn't use gradients directly).

 

[Mangala]

Wow, that was a terrible oversight on my part. I believe I took that from the other question I was looking at at the time. This gradient would be 2xi+2zk, no?

 

[n1person]

yeep! looks like it!