THE DIVERGENCE THEOREM | Improve Society

Let $S$ be a closed surface bounding a region of volume $V$ . Choose the outward drawn normal to the surface as the positive normal and assume that $\alpha,\ \beta,\ \gamma$ are the angles which this normal makes with the positive $x$ , $y$ and $z$ axes respectively. Then if $A_1,\ A_2$ and $A_3$ are continuous and have continuous partial derivatives in the region

$\displaystyle \underset{V}{\iiint}\left( \frac{\partial A_1}{\partial x} + \frac{\partial A_2}{\partial y} + \frac{\partial A_3}{\partial z} \right)dV = \underset{S}{\iint}(A_1\cos\alpha + A_2\cos\beta + A_3\cos\gamma)dS\cdots(35)$

which can also be written

$\displaystyle \underset{V}{\iiint}\left( \frac{\partial A_1}{\partial x} + \frac{\partial A_2}{\partial y} + \frac{\partial A_3}{\partial z} \right)dV = \underset{S}{\iint}[ A_1dydz + A_2dzdx + A_3dxdy ]\cdots (36)$

In vector form with $\bold{A} = A_1\bold{i} + A_2\bold{j} + A_3\bold{k}$ and $\bold{n} = \cos\alpha\bold{i} + \cos\beta\bold{j} + \cos\gamma\bold{k}$ , these can be simply written as

$\displaystyle \underset{V}{\iiint}\nabla\cdot\bold{A}dV = \underset{S}{\iint}\bold{A}\cdot\bold{n}dS\cdots(37)$

In words this theorem, called the divergence theorem or Green’s theorem in space, states that the surface is equal to the integral of the normal component of a vector $\bold{A}$ taken over a closed surface is equal to the integral of the divergence of $\bold{A}$ taken over the volume enclosed by the surface.