.. index:: integration, surface integrals, volume integrals 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. General Case ------------ We want to integrate a function $f$ over a $k$-manifold in $\R^n$, parametrized as: .. math:: \mathbf{\varphi}: \R^k \to \R^n\quad \mathbf{\varphi}(t_1, t_2, \dots, t_k) = \mat{\varphi_1(t_1, t_2, \dots, t_k)\cr \varphi_2(t_1, t_2, \dots, t_k)\cr \vdots \cr \varphi_n(t_1, t_2, \dots, t_k)\cr } then the integral of $f(x_1, x_2, \dots, x_n)$ over $\varphi$ is: .. math:: \int_M f(x_1, x_2, \dots, x_n)\,\d S = \int_{\R^n} f(\mathbf{\varphi}(t_1, t_2, \dots, t_k))\sqrt{\det{\bf G}}\,\d t_1\d t_2\cdots\d t_k where ${\bf G}$ is called a Gram matrix and ${\bf J}$ is a Jacobian: .. math:: ({\bf G})_{ij} = ({\bf J}^T{\bf J})_{ij} = J_{ik}J_{kj} = {\partial\varphi_k\over\partial t_i} {\partial\varphi_k\over\partial t_j} ({\bf J})_{ij} = {\partial\varphi_i\over\partial t_j} = \mat{ {\partial\mathbf{\varphi}\over\partial t_1} & {\partial\mathbf{\varphi}\over\partial t_2} & \cdots & {\partial\mathbf{\varphi}\over\partial t_k} \cr \vdots & \vdots & \vdots & \vdots \cr \vdots & \vdots & \vdots & \vdots \cr \vdots & \vdots & \vdots & \vdots \cr } The idea behind this comes from the fact that the volume of the $k$-dimensional parallelepiped spanned by the vectors .. math:: {\partial\mathbf{\varphi}\over\partial t_1}, \dots, {\partial\mathbf{\varphi}\over\partial t_k} is given by .. math:: V = \sqrt{\det {\bf J}^T{\bf J}} where ${\bf J}$ is an $n\times k$ matrix having those vectors as its column vectors. Example ~~~~~~~ Let's integrate a function $f(x, y, z)$ over the surface of a sphere in 3D (e.g. $k=2$ and $n=3$): .. math:: \mathbf{\varphi}(\theta, \phi) &= \mat{ r\sin\theta\cos\phi \cr r\sin\theta\sin\phi \cr r\cos\theta \cr } {\bf J} &= \mat{ -r\sin\theta\sin\phi & r\cos\theta\cos\phi \cr r\sin\theta\cos\phi & r\cos\theta\sin\phi \cr 0 & -r\sin\theta \cr } {\bf G} &= {\bf J}^T {\bf J} = \mat{ -r\sin\theta\sin\phi & r\sin\theta\cos\phi & 0 \cr r\cos\theta\cos\phi & r\cos\theta\sin\phi & -r\sin\theta \cr } \mat{ -r\sin\theta\sin\phi & r\cos\theta\cos\phi \cr r\sin\theta\cos\phi & r\cos\theta\sin\phi \cr 0 & -r\sin\theta \cr } = \mat { r^2\sin^2\theta & 0 \cr 0 & r^2 \cr } \det{\bf G} &= r^4\sin^2\theta \sqrt{\det{\bf G}} &= r^2\sin\theta \int_M f(x, y, z) \d S &= \int_{\R^n} f(r\sin\theta\cos\phi, r\sin\theta\sin\phi, r\cos\theta)\, r^2\sin\theta\,\d\theta\,\d\phi = &= \int_0^\pi\d\theta \int_0^{2\pi}\d\phi\, f(r\sin\theta\cos\phi, r\sin\theta\sin\phi, r\cos\theta)\, r^2\sin\theta Let's say we want to calculate the surface area of a sphere, so we set $f(x, y, z) = 1$ and get: .. math:: \int_M \d S = \int_0^\pi\d\theta \int_0^{2\pi}\d\phi\, r^2\sin\theta = 2\pi r^2\int_0^\pi\d\theta \sin\theta = 4\pi r^2 Special Cases ------------- .. index:: pair: volume; integration k = n ~~~~~ .. math:: \det{\bf G} = \det {\bf J}^R{\bf J} = (\det {\bf J})^2 \d S = |\det {\bf J}|\,\d t_1\,\d t_2\cdots\d t_k .. index:: pair: line; integration k = 1 ~~~~~ .. math:: \det{\bf G} = \det \left( \left({\d\varphi_1\over\d t}\right)^2+ \left({\d\varphi_2\over\d t}\right)^2+ \cdots \right) = \left|{\d\mathbf{\varphi}\over\d t}\right|^2 \d S = \left|{\d\mathbf{\varphi}\over\d t}\right|\,\d t .. index:: pair: surface; integration k = n - 1 ~~~~~~~~~ .. math:: \det{\bf G} = \det {\bf J}^R{\bf J} = =\det(\cdots)^2 + \det(\cdots)^2+\cdots+\det(\dots)^2 = =\left|\det\mat{ {\partial\mathbf{\varphi}\over\partial t_1} & {\partial\mathbf{\varphi}\over\partial t_2} & \cdots & {\partial\mathbf{\varphi}\over\partial t_k} & {\bf e}_1 \cr \vdots & \vdots & \vdots & \vdots & {\bf e}_2 \cr \vdots & \vdots & \vdots & \vdots & \vdots \cr \vdots & \vdots & \vdots & \vdots & {\bf e}_n \cr }\right|^2 \equiv |\omega_\varphi|^2 \d S = |\omega_\varphi|\,\d t_1\,\d t_2\cdots\d t_k $\omega_\varphi$ is a generalization of a vector cross product. The $\det(\cdots)$ 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.). .. index:: pair: surface; integration k = 2, n = 3 ~~~~~~~~~~~~ .. math:: \det{\bf G} = \left|{\partial\mathbf{\varphi}\over\partial t_1}\times {\partial\mathbf{\varphi}\over\partial t_2}\right|^2 \d S = \left|{\partial\mathbf{\varphi}\over\partial t_1}\times {\partial\mathbf{\varphi}\over\partial t_2}\right|\,\d t_1\,\d t_2 y = f(x, z) ~~~~~~~~~~~ .. math:: \det{\bf G} = 1 +\left({\partial f\over\partial x}\right)^2 +\left({\partial f\over\partial z}\right)^2 \d S = \sqrt{1 +\left({\partial f\over\partial x}\right)^2 +\left({\partial f\over\partial z}\right)^2 }\,\d x\,\d z in general for $x_j = f(x_1, x_2, \dots, x_n)$ we get: .. math:: \det{\bf G} = 1 +\left({\partial f\over\partial x_1}\right)^2 +\left({\partial f\over\partial x_2}\right)^2 +\cdots \d S = \sqrt{1 +\left({\partial f\over\partial x_1}\right)^2 +\left({\partial f\over\partial x_2}\right)^2 +\cdots }\,\d x_1\,\d x_2\cdots\d x_n The "$x_j$" term is missing in the sums above. .. index:: pair: implicit surface; integration Implicit Surface ~~~~~~~~~~~~~~~~ For a surface given implicitly by .. math:: F(x_1, x_2, ..., x_n) = 0 we get: .. math:: \d S = |\nabla F| \left|{\partial F\over\partial x_n}\right|\,\d x_1\cdots\d x_{n-1} .. index:: pair: orthogonal coordinates; integration Orthogonal Coordinates ~~~~~~~~~~~~~~~~~~~~~~ If the coordinate vectors are orthogonal to each other: .. math:: {\partial\mathbf{\varphi}\over\partial t_i} \cdot {\partial\mathbf{\varphi}\over\partial t_i} = 0 \quad\text{for $i\neq j$} we get: .. math:: \d S = \left|{\partial\mathbf{\varphi}\over\partial t_1}\right| \left|{\partial\mathbf{\varphi}\over\partial t_2}\right| \cdots \left|{\partial\mathbf{\varphi}\over\partial t_k}\right| \d t_1\cdots\d t_k Motivation ---------- Let the $k$-dimensional parallelepiped P be spanned by the vectors .. math:: {\partial\mathbf{\varphi}\over t_1}, \dots, {\partial\mathbf{\varphi}\over t_k} and let $\mathbf{J}$ is $n\times k$ matrix having these vectors as its column vectors. Then the area of P is .. math:: V = \sqrt{\det {\bf J}^T{\bf J}} so the definition of the integral over a manifold is just approximating the surface by infinitesimal parallelepipeds and integrating over them. Example ------- Let's calculate the total distance traveled by a body in 1D, whose position is given by $s(t)$: .. math:: l = \int_\gamma \d s = \int_{t_1}^{t_2}\left|{\d s\over \d t}\right| \d t = = \int_{t_1}^{t'}\left|{\d s\over \d t}\right| \d t + \int_{t'}^{t''}\left|{\d s\over \d t}\right| \d t + \cdots + \int_{t''''^{\cdots}}^{t_2}\left|{\d s\over \d t}\right| \d t = =|s(t')-s(t_1)|+|s(t'')-s(t')|+\cdots+|s(t_2)-s(t''''^{\cdots})| where $t'$, $t''$, ... are all the points at which $\left|{\d s\over \d t}\right|=0$, so each of the integrals in the above sum has either positive or negative integrand.