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

In summary, the conversation discusses the derivation of the formula for arc length and the attempt to use the same method to find the surface area formula. The formula for arc length involves taking the integral of the square root of 1 plus the derivative of the function squared. However, this method does not work for finding the surface area formula and a different approach is needed, possibly involving a surface integral.
  • #1
okkvlt
53
0
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 here's 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?
 
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1. How do I calculate the arc length in higher dimensions?

To calculate the arc length in higher dimensions, you can use the formula L = ∫√(1 + (dy/dx)^2)dx where dy/dx represents the derivative of the function. This formula can be extended to higher dimensions by using partial derivatives and integration over the corresponding variables.

2. Is there a general formula for finding surface area in higher dimensions?

Yes, there is a general formula for finding surface area in higher dimensions. It is given by S = ∫√(1 + (fx)^2 + (fy)^2) dA where fx and fy represent the partial derivatives of the function with respect to the corresponding variables and dA represents the differential area element.

3. Can the arc length and surface area formulas be applied to any higher dimensional shape?

Yes, the arc length and surface area formulas can be applied to any higher dimensional shape as long as the necessary variables and derivatives can be determined. However, for more complex shapes, the calculations may be more difficult or require numerical methods.

4. How do I interpret the results when generalizing to higher dimensions?

The results when generalizing to higher dimensions can be interpreted in a similar way as for lower dimensions. The arc length represents the length of a curve in higher dimensions, while the surface area represents the area of a surface in higher dimensions. These values can be used for further calculations or to gain a better understanding of the shape in question.

5. Are there any limitations when generalizing arc length and surface area to higher dimensions?

One limitation when generalizing arc length and surface area to higher dimensions is that the calculations may become more complex for certain shapes. Additionally, the formulas may not be applicable for highly irregular or discontinuous shapes. In these cases, numerical methods may be necessary to approximate the values.

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