Multidimensional Integrals

1.

One dimensional integrals of functions of two (or more) dimensions are known as ``line integrals''. One has, given f(x,y),

$$\int_{path} f(x,y) ds$$

where ds is the differential element of distance along the specified path

2.

Two dimensional integrals of functions of two dimensions are related to areas.

$$\int_a^b dx \left ( \int_c^d dy f(x,y) \right )$$

If f(x,y) were 1, then this would give the rectangular area bounded by $a\leq x \leq b$ and $c \leq y \leq d$.

3.

Three dimensional integrals of functions of 3 dimensions are related to volumes.

$$\int_a^b dx \left ( \int_c^d dy \left ( \int_e^f dz g(x,y,z) \right ) \right )$$

If g(x,y,z) were 1, then this would give the rectangular volume bounded by $a\leq x \leq b$ and $c \leq y \leq d$ and $e \leq y \leq f$.

4.

In two dimensions, one often can use polar coordinates instead of cartesian, i.e., $(r , \theta)$ rather than (x,y). The differential area element changes from dxdy to $rdrd\theta$.

$$x = r \cos \theta$$

and

$$y = r \sin \theta$$

, with $0 \le r \le \infty$ and $0 \le \theta \le 2 \pi$.

5.

In 3 dimensions, one often can use spherical polar coordinates instead of cartesian, i.e., $(r, \theta, \phi)$ rather than (x,y,z). The differential volume element changes from dxdydz to $r^2dr \sin \theta d \theta d\phi$.

$$x = r \sin \theta \cos \phi$$

$$y = r \sin \theta \sin \phi$$

$$x = r \cos \theta$$

with $0 \le r \le \infty$, $0 \le \theta \le \pi$ and $0 \le \phi \le 2 \pi$.