The Divergence Theorem (pages in this section may load slowly)   

Surface integrals and flux have a number of applications in electricity and magnetism, where they often occur in connection with the fundamental theorem of calculus. In this section, we explore the Divergence theorem, which involves flux integrals and is important in the study of electricity.

To begin with, let us suppose that W is the solid between two surface z = a( x,y) and z = b( x,y) over a region R in the xy-plane, with the additional assumption that a( x,y) £ b( x,y) over R and a( x,y) = b(x,y) if ( x,y) is on the boundary of R.

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The upper surface is parameterized by r( x,y) = á x,y,b( x,y) ñ where (x,y) is in the region R. Thus, its vector surface differential is

dS = ( rx×ry) dxdy = á -bx, -by,1 ñ dxdy

Likewise, the lower surface is parameterized by r(x,y) = á x, y, a(x,y) ñ , and its vector surface differential is

dS = ( ry×rx) dxdy = á ax, ay,-1 ñ dxdy

where the difference in signs is necessary for the surface normal to be pointing outside of the surface on both the bottom and the top.

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Suppose now that F( x,y,z) = á 0,0,P(x,y,z) ñ . Then the flux of F through the closed boundary surface of W is given by

¶W  F·dS = ó õ ó õ top  F·dS+ ó õ ó õ bottom  F·dS = ó õ ó õ R  á 0,0,P( x,y,b( x,y) ) ñ · á -bx,-by,1 ñ dxdy + ó õ ó õ R  á 0,0,P( x,y,a( x,y)) ñ · á ax,ay,-1 ñ dxdy = ó õ ó õ R  P( x,y,b( x,y) ) dxdy- ó õ ó õ R  P( x,y,a( x,y) ) dxdy = ó õ ó õ R  P( x,y,b( x,y) ) -P( x,y,a(x,y) ) dA

However, P( x,y,b) -P( x,y,a) = ò_a^bP_zdz, which implies that

¶W  F·dS = ó õ ó õ R  P( x,y,b( x,y) ) -P( x,y,a(x,y) ) dA = ó õ ó õ R  ó õ b( x,y) a( x,y)     ¶P ¶z dzdA = ó õ ó õ ó õ W      ¶P ¶z dV

Since div( F) = ¶_x0+¶_y0+¶_zP, we can write this result in terms of the vector field as

¶W  F·dS = ó õ ó õ ó õ W  div( F)   dV

(1)

Equation (1) is known as the divergence theorem. It allows us to calculate the flux of a vector field through a closed surface using a triple integral over the solid bounded by the surface.

       

EXAMPLE 1    Find the flux of F( x,y,z) = á0, 0, z ñ through the solid between z = 1 - x² - y² and z = 0.