# 3.1. Integration¶

This chapter doesn’t assume any knowledge about differential geometry. The most versatile way to do integration over manifolds is explained in the differential geometry section.

## 3.1.1. General Case¶

We want to integrate a function over a -manifold in , parametrized as:

then the integral of over is:

where is called a Gram matrix and is a Jacobian:

The idea behind this comes from the fact that the volume of the -dimensional parallelepiped spanned by the vectors

is given by

where is an matrix having those vectors as its column vectors.

### Example¶

Let’s integrate a function over the surface of a sphere in 3D (e.g. and ):

Let’s say we want to calculate the surface area of a sphere, so we set and get:

## 3.1.2. Special Cases¶

### k = n - 1¶

is a generalization of a vector cross product. The symbol means a determinant of a matrix with one row removed (first term in the sum has first row removed, second term has second row removed, etc.).

### y = f(x, z)¶

in general for we get:

The “” term is missing in the sums above.

### Implicit Surface¶

For a surface given implicitly by

we get:

### Orthogonal Coordinates¶

If the coordinate vectors are orthogonal to each other:

we get:

## 3.1.3. Motivation¶

Let the -dimensional parallelepiped P be spanned by the vectors

and let is matrix having these vectors as its column vectors. Then the area of P is

so the definition of the integral over a manifold is just approximating the surface by infinitesimal parallelepipeds and integrating over them.

## 3.1.4. Example¶

Let’s calculate the total distance traveled by a body in 1D, whose position is given by :

where , , … are all the points at which , so each of the integrals in the above sum has either positive or negative integrand.