= Gradient first moment
The <gradient first moment> of an effective <magnetic field gradient> waveform $G(t)$ is its time-weighted <integral>,
$$
m_1=\int tG(t)\,dt.
$$
Here effective includes the sign changes caused by refocusing pulses. Its units are $\mathrm{T\,s^2/m}$ when $G$ is in tesla per metre. The <gradient zeroth moment> is $m_0=\int G(t)\,dt$. For position $x(t)=x_0+vt$ with constant <velocity>, the accumulated <nuclear magnetic resonance> <phase> is
$$
\phi=\gamma\int G(t)x(t)\,dt=\gamma(x_0m_0+vm_1),
$$
where $\gamma$ is the angular <gyromagnetic ratio>. Thus $m_0=0$ removes the dependence on initial position while $m_1$ can retain sensitivity to <velocity>, the basis of an idealized <phase-contrast magnetic resonance imaging> encoding.<A flow velocity zeugmatographic interlace for NMR imaging in humans by P. R. Moran>[Original abstract; motion-sensitive phase encoding; equations here are an explicit constant-velocity derivation]{ref}
For a teaching <bipolar gradient>, let the first lobe have signed area $A$ and time centroid $t_1$, and the second have signed area $-A$ and centroid $t_2$. Then
$$
m_0=A-A=0,\qquad m_1=A(t_1-t_2).
$$
Equal and opposite lobe areas cancel stationary-spin phase, but their time separation prevents cancellation of the constant-velocity phase. For $A=10^{-5}\,\mathrm{T\,s/m}$ and $t_2-t_1=2\,\mathrm{ms}$, the first moment is $-2\times10^{-8}\,\mathrm{T\,s^2/m}$. These are illustrative waveform values, not measurements from the source.
The time origin is part of the definition: replacing $t$ by $t-t_0$ gives $m_1'=m_1-t_0m_0$. Balanced $m_0=0$ makes the first moment independent of that choice. Two encodings with equal zeroth moments use their first-moment difference $\Delta m_1$, rather than either individual value, to infer <velocity>.
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