page_title: Pi
section_title: Vector calculus
hierarchical_section_title: Pi / Role and characterizations in mathematics / Vector calculus
caption_attribution_description: Français&#160;: Y10,5 Harmonique sphérique + sphère Légende&#160;: Harmonique sphérique&#160;: exemple L=10;m=5&#160;: la figure indique la représentation de la surface F = 5 + Ylm&#160;: les parties claires sont plus éloignées du centre (0,0,0) que les parties sombres. Source&#160;: créée à l'occasion du développement de l'article harmoniques sphériques, image obtenue avec le logiciel Matlab(R). Statut&#160;: Licence de documentation libre GNU (GFDL&#160;: GNU Free Documentation Licence).,
context_page_description: The number π is a mathematical constant. It is defined as the ratio of a circle's circumference to its diameter, and it also has various equivalent definitions. It appears in many formulas in all areas of mathematics and physics. It is approximately equal to 3.14159. It has been represented by the Greek letter "π" since the mid-18th century, and is spelled out as "pi". It is also referred to as Archimedes' constant.
  Being an irrational number, π cannot be expressed as a common fraction, although fractions such as 22/7 are commonly used to approximate it. Equivalently, its decimal representation never ends and never settles into a permanently repeating pattern. Its decimal digits appear to be randomly distributed, and are conjectured to satisfy a specific kind of statistical randomness.
  It is known that π is a transcendental number: it is not the root of any polynomial with rational coefficients. The transcendence of π implies that it is impossible to solve the ancient challenge of squaring the circle with a compass and straightedge.
  Ancient civilizations, including the Egyptians and Babylonians, required fairly accurate approximations of π for practical computations.
context_section_description: Vector calculus is a branch of calculus that is concerned with the properties of vector fields, and has many physical applications such as to electricity and magnetism.  The Newtonian potential for a point source Q situated at the origin of a three-dimensional Cartesian coordinate system is
  which represents the potential energy of a unit mass (or charge) placed a distance |x| from the source, and k is a dimensional constant.  The field, denoted here by E, which may be the (Newtonian) gravitational field or the (Coulomb) electric field, is the negative gradient of the potential:
  Special cases include Coulomb's law and Newton's law of universal gravitation. Gauss' law states that the outward flux of the field through any smooth, simple, closed, orientable surface S containing the origin is equal to 4πkQ:
    
  It is standard to absorb this factor of 4π into the constant k, but this argument shows why it must appear somewhere.  Furthermore, 4π is the surface area of the unit sphere, but we have not assumed that S is the sphere.  However, as a consequence of the divergence theorem, because the region away from the origin is vacuum (source-free) it is only the homology class of the surface S in R³\{0}  that matters in computing the integral, so it can be replaced by any convenient surface in the same homology class, in particular, a sphere, where spherical coordinates can be used to calculate the integral.
  A consequence of the Gauss law is that the negative Laplacian of the potential V is equal to 4πkQ times the Dirac delta function:
  More general distributions of matter (or charge) are obtained from this by convolution, giving the Poisson equation
  where ρ is the distribution function.
  The constant π also plays an analogous role in four-dimensional potentials associated with Einstein's equations, a fundamental formula which forms the basis of the general theory of relativity and describes the fundamental interaction of gravitation as a result of spacetime being curved by matter and energy:
  where Rμν is the Ricci curvature tensor, R is the scalar curvature, gμν is the metric tensor, Λ is the cosmological constant, G is Newton's gravitational constant, c is the speed of light in vacuum, and Tμν is the stress–energy tensor.  The left-hand side of Einstein's equation is a non-linear analogue of the Laplacian of the metric tensor, and reduces to that in the weak field limit, with the  term playing the role of a Lagrange multiplier, and the right-hand side is the analogue of the distribution function, times 8π.
caption: The techniques of vector calculus can be understood in terms of decompositions into spherical harmonics (shown)
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