# Multivariate functions

## Multivariate space

### Regions

A region is a subset

#### Elementary regions

An elementary region is a region which is either a type-I region or a type-II region.

#### Simple regions

A simple region is a region which is both a type-I and a type-II region.

### Curves and closed curves

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.

#### Closed curves

If the start and end point of the curve are the same then the curve is closed. We can write this as:

$$\oint_C f(r) ds =\int_a^b f(r(t)) |r'(t)| dt$$

### Length of a curve

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:

$$L=\lim_{n\rightarrow \infty }\sum_{i=1}^n ||f(t_{i})-f(t_{i-1})||$$

#### Method 1

$$L=\lim_{n\rightarrow \infty }\sum_{i=1}^n ||f(t_{i})-f(t_{i-1})||$$

$$L=\lim_{n\rightarrow \infty }\sum_{i=1}^n ||\dfrac{f(t_{i})-f(t_{i-1})}{\Delta t}||\Delta t$$

$$L=\lim_{n\rightarrow \infty }\sum_{i=1}^n ||f'(t)||\Delta t$$

$$L=\int_a^b ||f'(t)||dt$$

#### Method 2

$$L=\lim_{n\rightarrow \infty }\sum_{i=1}^n ||f(t_{i})-f(t_{i-1})||$$

$$L=\lim_{n\rightarrow \infty }\sum_{i=1}^n \sqrt {(f(t_{i})-f(t_{i-1}))^*M(f(t_{i})-f(t_{i-1}))}$$

$$L=\int_a^b \sqrt {(dt)^TM(dt)}$$