Projection / capability. Choose a boundary and reconcile storage, flux, sources/sinks, local continuity, and entropy production.
Question this mathematics answers. When the amount inside changes, did it cross the boundary, appear through a source, disappear through a sink, or get mis-accounted?
Equation family
Global balance: d/dt ∫V ρ dV = −∮∂V J·n dA + ∫V s dVLocal continuity: ∂ρ/∂t + ∇·J = sEntropy balance: ∂s/∂t + ∇·J_s = σ_s, with σ_s≥0 for the modeled irreversible production
Symbols. ρ = stored density; V = chosen region; J = flux; n = outward normal; s = source density; J_s = entropy flux; σ_s = entropy production density.
Assumptions. Fields are sufficiently regular for the integral/local conversion; the meaning and units of ρ,J,s depend on the domain. Entropy relation uses the declared thermodynamic/continuum model.
Where the inputs come from. Storage comes from inventory/state measurements; flux from boundary flow measurements or model; sources/sinks from processes inside the boundary. The boundary is a modeling/measurement choice that must be declared.
Short derivation / justification
Use the divergence theorem: ∮J·n dA=∫∇·J dV. If the global balance holds for arbitrary regions, the integrands yield ∂tρ+∇·J=s.
Worked example
Coolant: inflow 3, outflow 5, internal source 1 units/min ⇒ storage rate 3−5+1=−1 unit/min. Expand the boundary to include the receiver and what looked like external loss may become internal transfer.
Cadet or real-world application. Mass/energy/charge balances, inventory, thermal management, resource tracking.
What to look for. A local quantity can decrease even with nonnegative internal production if export through the boundary dominates.
Common misinterpretation. “If it disappeared from my chosen region it was destroyed.”
Break / domain boundary. The balance grammar transfers; the physical nouns, constitutive laws, units, and ontology do not automatically transfer.