# Unit Normal Vector to a Cylinder

#### Mangala

1. The problem statement, all variables and given/known data
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.

2. Relevant equations
$$\nabla$$F=Fxi+Fyj+Fzk

3. 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

Homework Helper
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!

### The Physics Forums Way

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving