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.

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.

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:

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.).

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.

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.