How do you calculate a vector area?

[Jake110]

this may not be the right section so sorry if its in the wrong place.


I've tried researching this but I find it's pretty hard to get information on obscure subjects. I talked to my teachers at college but they only teach A level so they didn't have a clue.

I've found some http://farside.ph.utexas.edu/teaching/316/lectures/node4.html" on vector areas on the internet but it's kind of confusing, could someone help me out and show me how to calculate it for a 3D object.

I need it for part of a much larger and more complicated set of physics equations, I've spent the last two months putting it all together and this is the last piece of the puzzle. the object I need to calculate the vector area of is an air core solenoid, ill attach a picture of the general shape.

 

[LCKurtz]

Not sure what level you are or if you are asking for what I think you are asking. Generally a surface can be given as a vector function of two variables:

$$\vec{R}(u,v) = \langle x(u,v), y(u,v), z(u,v) \rangle,\ (u,v)\in D$$

The vector element of surface are would be

$$d\vec S = \vec R_u \times \vec R_v dudv$$

and I'm guessing what you want is

$$\int\int_D d\vec S$$

 

[Jake110]

thanks for the reply but i think the equation i need the vector area for is wrong. plus I'm only at A level so i don't really understand that equation.

EDIT: unless its not the vector area of the entire object but just the cross sectional area, i think i just need to calculate the vector area of a disk. could you explain a bit more on how to use the equation you posted?

 

[LCKurtz]

thanks for the reply but i think the equation i need the vector area for is wrong. plus I'm only at A level so i don't really understand that equation.

EDIT: unless its not the vector area of the entire object but just the cross sectional area, i think i just need to calculate the vector area of a disk. could you explain a bit more on how to use the equation you posted?

I don't know what the "A level" is. You pretty much need Calc III to talk about general surface areas. The typical application of vector surface elements is in the calculation of flux integrals. These have the form

$$\int \int_S\vec{F} \cdot d\vec S = \int\int_D\vec{F}(u,v) \cdot \vec {R}_u(u,v) \times \vec{R}_v(u,v)\, dudv$$

and are used to calculate flux through an oriented surface.

If you just want the vector area of a disk that would be a vector perpendicular to the disk whose length is numerically equal to the area of the disk. Hope that helps.

 

[Jake110]

sorry, A Level is a two year program involving 3-5 subjects for students 17-18 years old, its what we have here in England. I'm 18 so what ever the equivalent level of education is in your country.

anyway, I talked to a guy on another forum and it seems to find the vector area of a 2D object is the same as the area of that object. so the shape i think i need is the same as a CD, a disk with a circular hole in the center. that means i just need to get the area of the main circle and minus the area of the hole in the center, the direction being perpendicular to the plane.

you seem to know more about it so i thought I'd ask you if that made sense or has the other guy got it wrong? it just seems to simple to me.

EDIT: ok i read your last post again and i guess that is how to do it, the area of the flat surface equals the length of the vector. awesome, thank you for your help.