Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

How do i generalize this result to higher dimensions? (arc length, surface area)

  1. Dec 31, 2009 #1
    a derivation of the formula for arc length is simple enough:
    given a function f[x], find the length of the arc from x0 to x1.

    lim(x1-x0)/n=dx
    n->inf

    x1
    [tex]S=^{i=n-1}_{i=0}\sum\sqrt{(x+(i+1)dx-(x+idx))^2+f(x+(i+1)dx)-f(x+dx))^2}[/tex]
    xo
    [tex]S=^{i=n-1}_{i=0}\sum\sqrt{(dx)^2+f(x+(i+1)dx)-f(x+idx))^2}[/tex]
    by the definition of the derivative, [tex]f(x+(i+1)dx)-f(x+idx)=f'(x+idx)*dx[/tex]
    [tex]S=^{i=n-1}_{i=0}\sum\sqrt{dx^2+(f'(x+idx)*dx)^2}[/tex]
    [tex]S=^{i=n-1}_{i=0}\sum dx*\sqrt{1+f'(x+idx)^2}[/tex]
    and by the definition of the integral
    [tex]^{i=n-1}_{i=0}\sum dx*\sqrt{1+f'(x+idx)^2}=\int\sqrt{1+f'(x)^2}dx[/tex]

    (the first equation uses the pythagorean theorem to estimate the length of the curve from x+idx,f(x+idx) to x+(i+1)dx,f(x+(i+1)dx).
    now heres where it gets messed up. suppose i want to find the surface area of the function f[x,y] by the same technique.
    i have a square,

    D_______C
    |...........|
    |...........|
    |_______|
    A...........B

    [tex]A=x+idx,y+jdy[/tex]
    [tex]B=x+(i+1)dx,y+jdy[/tex]
    [tex]C=x+(i+1)dx,y+(j+1)dy[/tex]
    [tex]D=x+idx,y+(j+1)dy [/tex]
    where
    lim(x1-x0)/nx=dx
    nx->inf
    lim(y1-y0)/ny=dy
    ny->inf

    anyway, to avoid a long drawn out thing that arrives to the wrong conclusion, i multiplied the distance A,f(A) to B,f(B) by the distance A,f(A) to D,f(D) and i came up with the integrand being
    [tex]\sqrt{1+(\partial f/\partial x)^2+(\partial f/\partial y)^2+(\partial f/\partial y)(\partial f/\partial x)}[/tex]

    which is wrong. How do i use the same method of finding the arc length formula to find the surface area formula?
     
  2. jcsd
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted