bibkey: openstax2016boundaryintegrals authors: Gilbert Strang and Edwin Herman year: 2016 title: Calculus Volume 3 doi: null url: https://openstax.org/books/calculus-volume-3 claim: Green’s theorem and the outward oriented divergence theorem supply the classical boundary integral ingredients for planar area and three dimensional volume. strata_touched: [] license: citation-only triage: anchor

Calculus Volume 3

OpenStax, Rice University. Section 6.4 states Green’s theorem for a positively oriented boundary and a vector field with the required continuous derivatives. With $(P,Q)=(-y/2,x/2)$ it gives $A=\frac{1}{2}\oint (xdy-ydx)$. Integration along a straight edge from $a$ to $b$ gives $\frac{1}{2}\det (a,b)$.

Section 6.8, equation (6.24), states the divergence theorem for a closed surface with outward orientation. Applying it to $F(x)=x/3$ gives $V=\frac{1}{3}{\int }_{\partial K}x\cdot ndS$. For an outward oriented triangle $(a,b,c)$ its contribution is $\det (a,b,c)/6$. The same theorem on a ball of radius $R$ yields $V=RS/3$.

These are classical area and volume ingredients. Reversing all face orientations changes the sign of the determinant sum. A scalar area or volume is not a complete boundary record for an arbitrary later task.