# 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 by a Lipschitz continuous real-valued function of defined on the unit square , that is .

We define by iterated 1D integration

- ,

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

- .

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

- .

*Cauchy’s inequality* takes the form

- .

# 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 is a volume with boundary with outward unit normal , then

- ,

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:

- .

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

- .

# Learn More

- \hyperref[chaptergaussgreen]{The Divergence Theorem}

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

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