Q:

questions regarding definition of bounded variation of several variables

$\int_\Omega u(x)\,\mathrm{div}\boldsymbol{\phi}(x)\mathrm{d}x = -\int_\Omega \langle\boldsymbol{\phi}, Du(x)\rangle \qquad \forall\boldsymbol{\phi}\in C_c^1(\Omega,\mathbb{R}^n)$ (http://en.wikipedia.org/wiki/Bounded_variation#BV_functions_of_several_variables)

I am not sure what this means, and why this equality is being established in setting definition of BV functions of several variable. Can anyone explain this? Also, what is $\langle\boldsymbol{\phi}, Du(x)\rangle$ referring to?

A:

$\langle\boldsymbol{\phi}, Du(x)\rangle$ is the scalar product of two vectors, one of which is the gradient of $u$, the other is the vector field $\phi$. The first part would be integration by parts if $u$ had gradient $Du$ in the classical sense. Since we do not assume that $Du$ exists in the classical sense, the equation you quoted is taken as the definition of $Du$. A vector-valued measure $Du$ that satisfies the equation for all smooth "test" functions $\phi$ is called the distributional derivative of $Du$.  
As long as you are reading Wikipedia, you can consult its article on distributions. But generally, it's better to read a book. The definition of BV was recently discussed here, where I gave a reference to an accessible book on the subject. 

