I wonder about statement (1). It is given that u is a column vector, A and B are constant, symmetric, square matrices of such size that the expressions on the left hand side of (1) are well defined. It is also stated that $\hat{n}$ is the outward pointing unit normal to $\partial\Omega$, $\hat{x}$ and $\hat{y}$ are unit vectors in the x- and y-directions and ds is the infinitesimal arclength element counted clockwise around $\Omega$. $$(1) \,\,\, -\oint_{\partial \Omega}u^TA\,u\,dy + \oint_{\partial \Omega}u^TB\,u\,dx = -\oint_{\partial \Omega}u^T(A\hat{x} + B\hat{y})\,u\cdot \hat nds$$
Also, later in the text is $(A\hat{x} + B\hat{y})\cdot \hat n$ from the right hand side of (1) considered, as if $u^T(A\hat{x} + B\hat{y})\,u\cdot \hat n = u^T(A\hat{x} + B\hat{y})\cdot \hat n\,u$.
No further explanation is given in the text, but the results depend quite heavily on (1) and therefore do I assume that it is correct. But some information about notation or assumptions have been left out, since for me is the right hand side of (1) not well defined.. I would very much appreciate if someone could explain how (1) is justified, since I just end up with non-defined matrix multiplications due to different dimensions.