A region is a subset
An elementary region is a region which is either a type-I region or a type-II region.
A simple region is a region which is both a type-I and a type-II region.
In a space we can identify a curve between two points. If the input in the real numbers then this curve is unique.
For more general scalar fields this will not be the case. Two points in \(\mathbb{R}^2\) could be joined by an infinite number of paths.
A curve can be defined as a function on the real numbers. The curve itself is totally ordered, and homogenous to the real number line.
We can write the curve therefore as:
\(r:[a,b]\rightarrow C\)
Where \(a\) and \(b\) are the start and end points of the curve, and \(C\) is the resulting curve.
If the start and end point of the curve are the same then the curve is closed. We can write this as:
\(\oint_C f(r) ds =\int_a^b f(r(t)) |r'(t)| dt\)
We have a curve from \(a\) to \(b\) in \(\mathbf R^n\).
\(f:[a,b]\rightarrow \mathbf R^n\)
We divide this into \(n\) segments.
The \(i\)th cut is at:
\(t_i=a+\dfrac{i}{n}(b-a)\)
So the first cut is at:
\(t_0=a\)
\(t_n=b\)
The distance between two sequential cuts is:
\(||f(t_i)-f(t_{i-1}||\)
The sum of all these differences is:
\(L=\sum_{i=1}^n ||f(t_{i})-f(t_{i-1})||\)
The limit is:
\(L=\lim_{n\rightarrow \infty }\sum_{i=1}^n ||f(t_{i})-f(t_{i-1})||\)
\(L=\lim_{n\rightarrow \infty }\sum_{i=1}^n ||f(t_{i})-f(t_{i-1})||\)
\(L=\lim_{n\rightarrow \infty }\sum_{i=1}^n ||\dfrac{f(t_{i})-f(t_{i-1})}{\Delta t}||\Delta t\)
\(L=\lim_{n\rightarrow \infty }\sum_{i=1}^n ||f'(t)||\Delta t\)
\(L=\int_a^b ||f'(t)||dt\)
\(L=\lim_{n\rightarrow \infty }\sum_{i=1}^n ||f(t_{i})-f(t_{i-1})||\)
\(L=\lim_{n\rightarrow \infty }\sum_{i=1}^n \sqrt {(f(t_{i})-f(t_{i-1}))^*M(f(t_{i})-f(t_{i-1}))}\)
\(L=\int_a^b \sqrt {(dt)^TM(dt)}\)