= Magnetic helicity
{title2=$H=\int_V\mathbf A\cdot\mathbf B\,dV,\quad\mathbf B=\nabla\times\mathbf A$}
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The <magnetic helicity> integrates the <dot product> of a <magnetic vector potential> and its <magnetic field>. It measures linkage and twisting of magnetic flux. For a <solenoidal vector field> $\mathbf B$ tangent to the boundary, it is unchanged by the <gauge transformation> $\mathbf A\mapsto\mathbf A+\nabla\chi$, since the change is $\int_V\nabla\cdot(\chi\mathbf B)\,dV=\int_{\partial V}\chi\mathbf B\cdot d\mathbf S=0$. If its density satisfies $\partial_t h=\nabla\cdot(f\mathbf B-h\mathbf u)$ with both vector fields tangent to the boundary of a fixed <volume>, the <divergence theorem> gives conservation of its integral. This is an instance of <helicity conservation by tangent boundary conditions>.
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