yan... 發問於 科學及數學數學 · 1 十年前

surface integral & surface area

i would like to know the relationship between surface integral and surface area.

how are they actually related???

Thanks a lot!!!!!


plz give some examples~~~ and it would be great if u can explain more clearly with the formulas~~

1 個解答

  • 1 十年前

    In 3 dimensions, You can have two type of surface areas

    1. it bounds a volume which means it is closed.

    To work out the Surface area(S. A.), you need to know the shape of the volume and all the planes on the volume

    e.g. a cube is bounded by 6 surfaces and you know the way to work out the area of each surface.

    e.g. a sphere. The formula is S.A. = 4*Pi*r^2

    2. it is open and is called a curve(plane).

    In this case, you need to know the regions of the S.A.

    If it is not a flat plan, it is always hard to be calculated

    and here it comes the Surface Integral

    Normally, we express a surface as a function of x and y, i.e. z=f(x,y)

    We would like to express it as a vector function:


    and we want to have 2 variables instead of 3, so let x=u, y=v

    then z=f(u,v)

    and r becomes

    r=ui+vj+f(u,v)k (note that x=u, y=v is not necessary, it can be in other forms)

    then we differentiate r with respect to u and v

    and we get ru and rv

    |ru x rv| is a parallelogram

    This area approximates the area of a small part of the surface

    Adding these small areas(In the sense of the Riemann Integral)

    We get the area of the surface:

    A(s)=int(int(|ru x rv|)du)dv in a Domain which is the region of the surface

    Besides working out S.A., Surface Integral can work out the integration of a function on a plain.

    The formula will become int(G(x,y,z))ds where G is the function we want to integrate and s is the surface we want to integrate on.

    It can be written as I = int(int( G( x(u,v),y(u,v),z(u,v) )*|ru x rv| )du)dv

    I hope that is enough for you.

    It is quite hard for you to learn it on your own, so I suggest you wait until you get a place in university.

    Good luck

    資料來源: My lecture notes