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A real triplet with a rotationally invariant common mass term has three conserved internal currents. With the displayed sign convention, the current for $\delta \mathbf{\varphi }=\theta \mathbf{n}\times \mathbf{\varphi }$ is the negative of the conventional canonical current contracted with $\mathbf{n}$. Overall current sign does not affect conservation. Integrated charges require convergent integrals and vanishing net boundary flux; a local continuity equation alone does not guarantee conservation in an open region.

For an energy-conserving strong spherical shock wave expanding into mass density $C{r}^{-\beta }$, $0\le \beta <3$, dimensional analysis gives the displayed radius law. Neglect ambient pressure, radiation losses and gravity in this blast-dominated approximation. Write $u=\dot{R}U(\xi )$, $\rho =C{R}^{-\beta }D(\xi )$ and $p=C{R}^{-\beta }{\dot{R}}^{2}P(\xi )$, with $\xi =r/R$. The radial continuity equation, momentum equation and adiabatic pressure equation reduce toThe Strong-shock Rankine-Hugoniot conditions give $U(1)=P(1)=2/(\gamma +1)$ and $D(1)=(\gamma +1)/(\gamma -1)$. The explosion energy fixes the radius coefficient through $E=4\pi C{R}^{3-\beta }{\dot{R}}^2{\int }_0^1[D{U}^2/2+P/(\gamma -1)]{\xi }^{2}d\xi$.

For a stationary, $z$-independent ideal magnetohydrodynamics flow with $E_z=0$, nonzero horizontal magnetic field and nonzero magnetohydrodynamic mass loading $k$, the displayed decomposition has $\mathbf{B}\cdot \nabla k=\mathbf{B}\cdot \nabla U=0$. The continuity equation gives the first invariant, and the ideal magnetohydrodynamic induction equation gives the second. Entropy and $V=u_z-B_z/({\mu }_{0}k)$ are also constant on each magnetic field line. Stationarity alone permits constant nonzero $E_z$, so it does not imply this decomposition.

Assume zero toroidal loop voltage in smooth steady ideal magnetohydrodynamics, so the poloidal flow aligns with the poloidal magnetic field. Write $\rho {\mathbf{u}}_p=\eta {\mathbf{B}}_p$. The continuity equation makes $\eta$ constant along a magnetic line. The induction equation gives the conserved field-line angular velocity ${\Omega }_F=\Omega -\eta B_{\varphi }/(\rho R)$. The azimuthal momentum equation is $\rho {\mathbf{u}}_p\cdot \nabla (R^2\Omega )={\mathbf{B}}_p\cdot \nabla (R{B}_{\varphi })/{\mu }_0$; hence $L=R^2\Omega -R{B}_{\varphi }/({\mu }_0\eta )$ is also constant along the line.

With $M_{Ap}^2={\mu }_0{\eta }^2/\rho =|{\mathbf{u}}_p{|}^2/v_{Ap}^2$, eliminating $B_{\varphi }$ givesFor $M_{Ap}^2\ll 1$ and nonsingular invariants, $\Omega ={\Omega }_F+O(M_{Ap}^2)$ and matter approximately corotates along a field line. With exactly zero poloidal flow, induction directly gives Ferraro's law of isorotation. The approximation requires the correction $M_{Ap}^2({\Omega }_F-L/R^2)$ to be small on the segment in question.

Vector calculus studies differentiation and integration of scalar fields and vector fields. Three operators answer different local questions: the gradient describes change of a scalar value, the divergence describes net outward flow, and the curl describes local circulation. Their associated integral theorems turn local derivatives into measurements along paths and across boundaries.

The difference between divergence and curl is especially visible in two smooth vector fields on three-dimensional Euclidean space:At each point they have the same Euclidean norm $\sqrt{x^2+y^2}$, but their directions differ. For $F$, $\nabla \cdot F=2$ and $\nabla \times F=0$. For $G$, $\nabla \cdot G=0$ and $\nabla \times G=(0,0,2)$. One expands radially in each horizontal plane; the other describes rigid rotation.

Around the counterclockwise unit circle in the plane $z=0$, $F$ is normal to the boundary and has zero tangential component. Its outward planar flux integral is $2\pi$, while its line integral around the circle is zero. For $G$ the values are reversed: outward planar flux is zero, and circulation is $2\pi$. The divergence and curl predict exactly which boundary measurement survives.

Why is this useful? A velocity field can move material through a boundary or circulate it within a region. Conservation of mass connects its divergence with density changes through the continuity equation; its curl is vorticity. In electromagnetism, the Maxwell equations use the same operators to connect electric fields, magnetic fields, and their sources. The common mathematical language lets the same integral calculations work across different physical models. ^[ref].

The geometric framework behind these operators is developed in differential forms. The Generalized Stokes theorem unifies the fundamental theorem for line integrals, Green theorem, Stokes theorem, and divergence theorem. In coordinates their formulas look different, but each relates accumulation over a region to a measurement on its boundary.

Wilson's Vector Analysis, published in 1901, was based on J. Willard Gibbs's lectures at Yale. Wilson's preface explains that examples from geometry, mechanics, and physics are interspersed with the mathematical development, so readers encounter applications as soon as they have the tools to understand them. That is a useful way to approach vector calculus: learn each operation through a measurement it makes possible. ^[ref].

The void fraction is the fraction of a mixture's local volume occupied by gas. For a horizontally uniform bubbly liquid it is the transported concentration in a one-dimensional gas-volume continuity equation. It is distinct from gas mass fraction because gas and liquid have different mass densities.