{"id":5784,"date":"2014-10-06T06:05:18","date_gmt":"2014-10-05T21:05:18","guid":{"rendered":"http:\/\/fujiitoshiki.com\/improvesociety\/?p=5784"},"modified":"2014-08-08T19:41:50","modified_gmt":"2014-08-08T10:41:50","slug":"the-divergence-theorem","status":"publish","type":"post","link":"https:\/\/www.fujiitoshiki.com\/improvesociety\/?p=5784","title":{"rendered":"THE DIVERGENCE THEOREM"},"content":{"rendered":"
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Let S be a closed surface bounding a region of volume V. Choose the outward drawn normal to the surface as the positive normal<\/em> 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 <\/p>\n

\\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)<\/p>\n

which can also be written<\/p>\n

\\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)<\/p>\n

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 <\/p>\n

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

In words this theorem, called the divergence theorem<\/em> or Green’s theorem in space<\/em>, 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. <\/p>\n<\/blockquote>\n