Normal and Tangential Component of Vector on a Surface

[Legolaz]

Hello, given the figure above, how do I get the tangential and normal components of a vector in any plane by integration?

 

[SteamKing]

Hello, given the figure above, how do I get the tangential and normal components of a vector in any plane by integration?

You don't, at least not by integration.

To find the normal vector to a surface, you need to use the gradient:

http://math.oregonstate.edu/home/programs/undergrad/CalculusQuestStudyGuides/vcalc/grad/grad.html

http://mathworld.wolfram.com/Gradient.html

 

[Legolaz]

You don't, at least not by integration.

To find the normal vector to a surface, you need to use the gradient:

http://math.oregonstate.edu/home/programs/undergrad/CalculusQuestStudyGuides/vcalc/grad/grad.html

http://mathworld.wolfram.com/Gradient.html

Thank you for the reply, Steamking.

Say, I got now the the normal equation for the surface, my tangential would be the 2nd derivative of the gradient function, right?

My next problem is, I want to sum up all Normal and Tangential vectors on the surface, so that I may find the net or resultant vector acting upon the centroid/center of mass of the surface, how to do it?

 

[SteamKing]

Thank you for the reply, Steamking.

Say, I got now the the normal equation for the surface, my tangential would be the 2nd derivative of the gradient function, right?

No, you're missing something here. Normals and tangents to surfaces are a little different from normals and tangents to 2-D curves.

By evaluating the gradient of a surface function at a point, we are actually calculating a normal vector to the surface at that point. By using the normal vector and the gradient, the equation of the tangent plane at this point can also be determined. Tangent vectors at the same point are a little harder to pin down, unless a direction for the tangent is also specified.

http://math.kennesaw.edu/~plaval/math2203/gradient.pdf

My next problem is, I want to sum up all Normal and Tangential vectors on the surface, so that I may find the net or resultant vector acting upon the centroid/center of mass of the surface, how to do it?

Why? The resultant vector of what?

Have you studied any of the theorems of vector calculus yet? Gauss, Green, Stokes?

 

[Legolaz]

The resultant vector of all the tangent and the normal on the surface.

Nope, not yet.

 

[Mark44]

]My next problem is, I want to sum up all Normal and Tangential vectors on the surface, so that I may find the net or resultant vector acting upon the centroid/center of mass of the surface, how to do it?

Why? The resultant vector of what?

@Legolaz, the image you showed is just a 3-D figure defined by some surfaces. It makes no sense to sum vectors that are perpendicular to each of the bounding surfaces. For a resultant vector, you need to be working with forces, which are not mentioned so far in this thread.

 

[Legolaz]

@Legolaz, the image you showed is just a 3-D figure defined by some surfaces. It makes no sense to sum vectors that are perpendicular to each of the bounding surfaces. For a resultant vector, you need to be working with forces, which are not mentioned so far in this thread.

Yes Mark44, I understood and assume the ones I generally mentioned as "vectors" is referring to Force and Velocity gradients.

I mean, summing up vectors normal and tangential to the surface.