Finding a unit normal to a surface

In summary, on a surface with a curved boundary, you can find the unit normal by taking the vector product of the partial derivatives of the equation of the surface with respect to the two coordinates that are on the curve.
  • #1
dyn
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Hi.
I'm trying to self-study vector calculus and I have to admit I struggle with it. As regards finding normals to a surface I know 2 ways - one involves writing the surface equation as f(x.y.z) = 0 and taking the gradient. The other method involves writing the surface in terms of 2 parameters and taking the vector product of the 2 partial derivatives. Finally, my question. If I have a circle such as x2 + y2 = a2 I know the unit normal is k , but how do I show this ? My 2 methods don't work !
Thanks
 
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  • #2
The circle equation is the boundary of the surface. (## x^2+y^2=a^2 ## is the equation for a cylinder.) The actual equation of the plane in which the circle ## x^2+y^2=a^2 ## resides is simply ## z=0 ##. (I'm presuming you are wanting your surface to be a flat circle with center at the origin in the x-y plane). What you have for a surface is the endface of the cylinder. Take the gradient of ## z=0 ## and you do get ## \hat{n}=\hat{k} ##.
 
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  • #3
Yes i am taking the surface as a flat circle in the x-y plane. To take the gradient I need the form f(x,y,z)=0. What do I take f as ?
 
  • #4
dyn said:
Yes i am taking the surface as a flat circle in the x-y plane. To take the gradient I need the form f(x,y,z)=0. What do I take f as ?
## f(x,y,z)=z ##. Comes from ## z=0 ## is your equation of ## f(x,y,z)=0 ##. ## \nabla z=\hat{k} ##. ## \\ ## If the plane was instead ##z=5 ##, ## f(x,y,z)=z-5 ##, and again ## \nabla (z-5)=\hat{k} ##.
 
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  • #5
Thanks for your help. On a similar topic ; is it possible to find the unit normal to the curved surface of a cylinder orientated along the z-axis in this way. I can't find a way to write this surface as f(x,y,z) = 0 but I can get an answer using my other method using the vector product
 
  • #6
dyn said:
Thanks for your help. On a similar topic ; is it possible to find the unit normal to the curved surface of a cylinder orientated along the z-axis in this way. I can't find a way to write this surface as f(x,y,z) = 0 but I can get an answer using my other method using the vector product
The equation of the cylinder is simply ## x^2+y^2=a^2 ##. Thereby ## f(x,y,z)=0 ## is ## x^2+y^2-a^2=0 ##.
 
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  • #7
Thanks again. Much appreciated.
 
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1. What is a unit normal to a surface?

A unit normal to a surface is a vector that is perpendicular to the surface at a specific point and has a magnitude of 1. It is used in mathematics and physics to calculate surface integrals and to determine the direction of maximum curvature of a surface.

2. How do you find a unit normal to a surface?

To find a unit normal to a surface, you can use the gradient vector of the surface function at a specific point. The gradient vector will be perpendicular to the surface and its magnitude can be normalized to 1 to obtain the unit normal vector.

3. Can a surface have more than one unit normal vector?

Yes, a surface can have multiple unit normal vectors at different points. This is because the direction of the unit normal vector is dependent on the orientation of the surface at that point.

4. What is the significance of finding a unit normal to a surface?

Finding a unit normal to a surface is important in various fields of study such as mathematics, physics, and engineering. It is used to calculate surface integrals, determine the direction of maximum curvature of a surface, and to solve problems involving vectors and surfaces.

5. Can a unit normal vector change along a surface?

Yes, the unit normal vector can change along a surface as the orientation of the surface changes. This means that the direction of the unit normal vector can vary at different points on the surface.

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