Integration in Several Dimensions

  • This is a tricky domain because, unlike simple arithmetic, to solve a calculus problem – and in particular to perform integration -you have to be smart about which integration technique should be used: integration by partial fractions, integration by parts, and so on. (Marvin Minsky)
  • Many who have had an opportunity of knowing any more about mathematics confuse it with arithmetic, and consider it an arid science. In reality, however, it is a science which requires a great amount of imagination. — Say what you know, do what you must, come what may. — It is impossible to be a mathematician without being a poet in soul. (Sophia Kovalevskaya)

Let f(x)=f(x_1,x_2) by a Lipschitz continuous real-valued function of x=(x_1,x_2) defined on the unit square \Omega=\{x:0\le x_1,x_2\le 1\}, that is f:\Omega\rightarrow R.

We define by iterated 1D integration

  • \int_\Omega f(x)\,dx=\int_0^1(\int_0^1f(x_1,x_2)dx_1)\,dx_2=\int_0^1(\int_0^1f(x_1,x_2)dx_2)\, dx_1 ,

where the order of 1D integration is irrelevant, since both integrals express the common Riemann sum

  • \sum_{i,j=1}^Nf(ih,jh)h^2.

Generalization to any domain in 2d or 3d is direct, by iterated 1d integration. Linearity properties follow directly from the Riemann sum representation. By the triangle inequality it follows that

  • \vert \int_\Omega f(x)\, dx\vert\le \int_\Omega \vert f(x)\vert\, dx.

Cauchy’s inequality takes the form

  • \vert \int_\Omega f(x)g(x)\, dx\vert\le (\int_\Omega f^2(x)\, dx)^{\frac{1}{2}} (\int_\Omega g^2(x)\, dx)^{\frac{1}{2}}.

Learn More

  • \hyperref[chapterdoubleintegral]{Double integrals}
  • \hyperref[chaptermultipleintegral]{Multiple integrals}

To Think About

  • Is there a Fundamental Theorem for integration in 3d?

The Divergence Theorem

If \Omega is a volume with boundary \Gamma with outward unit normal n, then

  • \int_\Omega \nabla\cdot u\, dx=\int_\Gamma u\cdot n\, ds,

which is referred to as the Divergence Theorem or alternatively Gauss’ Theorem. We can see this result as a multidimensional analog of the Fundamental Theorem of Calculus:

  • \int_0^1u^\prime (x)dx =\int_0^1\frac{du}{dx}dx=u(1)-u(0).

In the case $\Omega$ is the unit square in 2d or unit cube in 3d, the Divergence Theorem follows directly from the Fundamental Theorem by considering the special case

  • \int_\Omega \frac{\partial u_1}{\partial x_1}\, dx_1dx_2=\int_\Gamma u_1 n_1\, dx_2.

Sonya Kovalevskaya}{Sonya (Sophia) Kovalevskaya} (1850-1891) of Russian origin, full professor of mathematics at the University of Stockholm 1889-91, remembered by the Cauchy-Kovalevskaya theorem, first female professor in Northern Europe: “It is impossible to be a mathematician without being a poet in soul…Many who have had an opportunity of knowing any more about mathematics confuse it with arithmetic, and consider it an arid science. In reality, however, it is a science which requires a great amount of imagination…Say what you know, do what you must, come what may…It seems to me that the poet has only to perceive that which others do not perceive, to look deeper than others look. And the mathematician must do the same thing.. .”

Learn More

  • \hyperref[chaptergaussgreen]{The Divergence Theorem}

Next: Gauss’, Green’s and Stokes’ Theorems

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