タグ: Green’s theorem in space

THE DIVERGENCE THEOREM

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.