# Point heat steady versus transient > The three-dimensional delta function, $$\delta^{(3)}(\vec{x})$$, is a mathematical tool defined by its integral property, where integrating over a volume $$V$$ yields a result of 1 only if the origin ($$\vec{x}=0$$) is contained within $$V$$, and 0 otherwise. This function is utilized in heat conduction to describe the concentration of an extensive quantity, such as a heat source, precisely located at a single point $$\vec{x}\_0$$, where the source term $$\kappa$$ is given by $$\kappa(\vec{x})=K \delta^{(3)}\left(\vec{x}-\vec{x}\_0\right)$$ for a source releasing heat at rate $K$. Although using the delta function is typically a simplification, it yields reasonably good results when the source size is indistinguishable from a point relative to measurement precision. Furthermore, the simulation of these point heat sources reveals two fundamentally different physical behaviors: if the source is continuous (steady state), the system achieves equilibrium and the temperature distribution near the source is classically inversely proportional; however, if the source is an instantaneous pulse (unsteady state), the entire system remains transient as the heat spreads outwards as a thermal wave, causing the temperature at every point to rise, peak, and subsequently decay back to the ambient zero level as the heat fully dissipates. ### :paperclips:IllustraDemo {% embed url="" %} ### :paperclips:IllustraDemo hub {% embed url="" %}