section_path: Role and characterizations in mathematics/Vector calculus and physics
section_name: Vector calculus and physics
paragraph_text: Vector calculus and physics
  
  The constant π is ubiquitous in vector calculus and potential theory, for example in Coulomb's law, Gauss's law, Maxwell's equations, and even the Einstein field equations. Perhaps the simplest example of this is the two-dimensional Newtonian potential, representing the potential of a point source at the origin, whose associated field has unit outward flux through any smooth and oriented closed surface enclosing the source: Φ (x) = 1 2 π log | x |. {\displaystyle \Phi (\mathbf {x})={\frac {1}{2\pi }}\log |\mathbf {x} |.} The factor of 1 / 2 π {\displaystyle 1/2\pi } is necessary to ensure that Φ {\displaystyle \Phi } is the fundamental solution of the Poisson equation in R 2 {\displaystyle \mathbb {R} ^{2}}: Δ Φ = δ {\displaystyle \Delta \Phi =\delta } where δ {\displaystyle \delta } is the Dirac delta function.
  
  In higher dimensions, factors of π are present because of a normalization by the n-dimensional volume of the unit n sphere. For example, in three dimensions, the Newtonian potential is: Φ (x) = − 1 4 π | x |, {\displaystyle \Phi (\mathbf {x})=-{\frac {1}{4\pi |\mathbf {x} |}},} which has the 2-dimensional volume (i.e., the area) of the unit 2-sphere in the denominator.