# Multivariate integration of scalar fields

## Integration of scalar fields

### Surface integral of scalar fields

In a scalar field, the line integral of the gradient field is the difference between the value of the scalar field at the start and end points.

This generalises the fundamental theorem of calclulus.

### Green’s theorem

We have a curve $$C$$ on a plane.

Inside this is region $$D$$.

We have two functions: $$L(x,y)$$ and $$M(x,y)$$ defined on the region and curve.

$$\oint_C (L dx + M dy)=\int \int_D (\dfrac{\delta M}{\delta x}-\dfrac{\delta L}{\delta y})dx dy$$

### Differential forms

#### Type-I

For type-I, we can integrate over y, then integrate over x.

#### Type-II

For type-II, we can integrate over x, then integrate over y.