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Let $q$ be a real harmonic function in a semistrip, $g_0(x)=q(x,0)$, $g_l(x)=q(x,l)$ and $h(y)=q(0,y)$. Let $G_j,N_j$ be the negative-exponential Half-range Fourier transforms of $g_j$ and $q_y(x,j)$, and put $H(k)={\int }_0^l{e}^{ky}h(y)dy$, $W(k)={\int }_0^l{e}^{ky}{q}_x(0,y)dy$. The closed differential form $e^{-ikz}{q}_{z}dz$ has zero boundary integral. Integration by parts of tangential derivatives cancels all corner values and gives the displayed global relation, for $\mathrm{Im}k\le 0$.

// Target: analysis.bigb

A symplectic form is a closed differential form of degree two whose value on every tangent space is a nondegenerate alternating bilinear form. In canonical position-momentum coordinates it is ${\sum }_{i}d{q}_i\wedge d{p}_i$.

For a nonvanishing adjoint Higgs field, put $n=\Phi /|\Phi |$. If an oriented orthonormal internal basis satisfies $[e_a,e_b]=\kappa {\epsilon }_{abc}{e}_c$, the displayed tensor is gauge invariant. In a gauge with $n=e_3$, write $A_j=a_j^a{e}_a$; the transverse quadratic terms cancel, leaving ${\mathcal{F}}_{jk}={\partial }_j{a}_k^3-{\partial }_k{a}_j^3$. Hence it is a closed differential form wherever $\Phi \ne 0$. Under trace normalization of su(2) with $-\mathrm{tr}(XY)$, ${\kappa }^2=2$ and the commutator term has coefficient $1/2$. With $-2\mathrm{tr}(XY)$, its coefficient is one. Omitting this normalization factor generally destroys closure.

The trace of the curvature form of a connection is a global closed differential form of degree two. Under a frame change $e^{\prime }=eg$, the connection matrix and curvature change by $A^{\prime }=g^{-1}Ag+g^{-1}dg$ and ${\Theta }^{\prime }=g^{-1}\Theta g$. Thus trace is independent of frame. The Bianchi identity gives $d\Theta =\Theta \wedge A-A\wedge \Theta$. Taking the trace cancels the two terms: one-forms commute with two-forms in the graded sense, and renaming the matrix indices gives equality of their traces. Locally $\mathrm{tr}\Theta =d\mathrm{tr}A$, since the trace of $A\wedge A$ vanishes by pairing its off-diagonal terms. It is the curvature of the determinant connection.