4.14 The divergence Theorem
Let \(E\) be a simple solid region and let \(S\) be the boundary surface of \(E\) given with positive (outward) orientation. Let \(F\) be a vector field whose component functions have continuous partial derivatives on an open region that contains \(E\). Then \[\iint \limits _S F\cdot dS = \iiint \limits _E \dive F\hspace {0.1cm}dV\]
Find the flux of the vector field \(F =z\,\textbf {i} + y\,\textbf {j} + x\,\textbf {k}\) over the unit sphere.
Solution.
\(\dive \hspace {0.1cm} F = 0 + 1 + 0 = 1\)
\begin {align*} \iint \limits _S F\cdot dS & = \iiint \limits _Bdiv\hspace {0.1cm}F\hspace {0.1cm}dV\\\\ & = \iiint \limits _B dV\\\\ & = Vol(B)\\\\ & =\frac {4\pi }{3}\\ \end {align*}
Questions on this section
Stuck on something here? Ask below and it stays attached to this topic.