GAUSS, Carl Friedrich (1777-1855). German mathematician, usually
considered along with Aristotle and Newton to be one of the three greatest mathematicians of all
time. Made important contributions to algebra, analysis, geometry, number theory,
numerical analysis, probability, and statistics, as well as astronomy and physics. See
PRIME—prime-number theorem.
Gauss' formulas
(or Delambre's analogies).
Formulas stating
the relations between the sine (or cosine) of half of the sum (or difference) of two
angles of a spherical triangle and the other angle and the three sides. If the angles of
the triangle are A, B, and C, and the sides opposite these angles are a, b, and c,
respectively, then Gauss' formulas are:
Gauss'
fundamental theorem of electrostatics. The surface integral of the exterior normal
component of the electric intensity over any closed surface all of whose points are free
of charge is equal to 4ï times the
total charge enclosed by the surface. In the corresponding theorem for gravitational
matter, the constant is - 4ï.
Gauss'
mean-value theorem. Let u be a regular harmonic function in a region R. Let P be a point
in R, S a sphere with center at P and lying entirely (boundary and interior points) within
R, and A the area of S.
Gauss' proof of
the fundamental theorem of algebra. The first known proof of this theorem. A geometrical
proof consisting essentially of substituting a complex number, a + bi, for the unknown of
the equation, separating the real and imaginary parts of the result, and then showing that
the two resulting functions of a and b are zero for some pair of values of a and b.
Gaussian
distribution. Same as normal distribution. See under NORMAL.
Theorem of
Gauss. The famous theorem that the total curvature of a surface is a function of the
fundamental coefficients of the first order of the surface and their partial derivatives
of the first and second orders. See above, Gauss' equation.
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