Theorem_multivector_surface_integral.tex

%
% Copyright © 2018 Peeter Joot.  All Rights Reserved.
% Licenced as described in the file LICENSE under the root directory of this GIT repository.
%
\maketheorem{Fundamental theorem for surface integrals.}{thm:surfaceintegral:100}{
Given a
% piecewise-smooth
continuous and connected surface
 \( S = \Bx(u, v) \)
parameterized by \( u \in [u_0, u_1], v \in [v_0, v_1] \), multivector functions \( F(\Bx), G(\Bx) \) that are differentable over \( S \), and
an (bivector-valued) area element \( d^2 \Bx = d\Bx_1 \wedge d\Bx_2 = du dv\, \Bx_u \wedge \Bx_v \)
\begin{equation*}
\int_S F d^2\Bx \lrboldpartial G
= \ointclockwise_{\partial S} F d\Bx G,
\end{equation*}
where \( \partial S \) is the boundary of the surface \( S \).
} % theorem