Unit Normal Vector to a Cylinder

In summary, to find the unit normal vector to the cylinder x^2+z^2=16, you need to find the gradient of the surface equation F(x,y,z)=x^2+z^2-16=0, which is 2xi+2zk. Then, parametrize the surface and plug it into the gradient to get the equation for the vectors. Finally, divide by the norm to obtain the unit vector. Alternatively, you can parametrize the cylinder and take the derivative to normalize it.
  • #1
Mangala
2
0

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


[tex]\nabla[/tex]F=Fxi+Fyj+Fzk


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!
 
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  • #2
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?
 
  • #3
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).
 
  • #4
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?
 
  • #5
yeep! looks like it!
 

FAQ: Unit Normal Vector to a Cylinder

What is a unit normal vector to a cylinder?

A unit normal vector to a cylinder is a vector that is perpendicular to the surface of the cylinder at a specific point. It has a magnitude of 1 and is used to indicate the direction in which the surface is facing.

How is a unit normal vector to a cylinder calculated?

The unit normal vector to a cylinder is calculated by taking the cross product of the tangent vector along the cylinder's surface and the vector that points towards the center of the cylinder. This can also be calculated by taking the gradient of the function that defines the surface of the cylinder.

Why is the unit normal vector important in cylinder geometry?

The unit normal vector is important in cylinder geometry because it helps determine the orientation and direction of the surface of the cylinder. It is also used in calculations involving surface integrals and curvature of the cylinder.

How does the unit normal vector change along the surface of a cylinder?

Along the surface of a cylinder, the unit normal vector changes direction but maintains a constant magnitude of 1. This is because the cylinder has a constant radius and the surface is curved, causing the normal vector to vary in direction.

Can a cylinder have multiple unit normal vectors?

No, a cylinder can only have one unit normal vector at any given point. This is because the unit normal vector is unique and is determined by the surface of the cylinder at that point. However, different points on the cylinder's surface may have different unit normal vectors.

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