Several other experiments followed, with André-Marie Ampère, who in 1820 discovered that the magnetic field circulating in a closed-path was related to the current flowing through the perimeter of the path ; Carl Friedrich Gauss ; Jean-Baptiste Biot and Félix Savart, both of which in 1820 came up with the Biot-Savart Law giving an equation for the magnetic field from a current-carrying wire ; Michael Faraday, who in 1831 found that a time-varying magnetic flux through a loop of wire induced a voltage, and others finding further links between magnetism and electricity.

It is important to note that while the electric flux is not affected by charges that are not within the closed surface, the net electric field, E, in the Gauss ' Law equation, can be affected by charges that lie outside the closed surface.

While Gauss ' Law holds for all situations, it is only useful for " by hand " calculations when high degrees of symmetry exist in the electric field.

The introduction of simple auxiliary terms, due to C. F. Gauss ( Dioptrische Untersuchungen, Göttingen, 1841 ), named the focal lengths and focal planes, permits the determination of the image of any object for any system ( see lens ).

The year 1796 was most productive for both Gauss and number theory.

Gauss also made important contributions to number theory with his 1801 book Disquisitiones Arithmeticae ( Latin, Arithmetical Investigations ), which, among things, introduced the symbol ≡ for congruence and used it in a clean presentation of modular arithmetic, contained the first two proofs of the law of quadratic reciprocity, developed the theories of binary and ternary quadratic forms, stated the class number problem for them, and showed that a regular heptadecagon ( 17-sided polygon ) can be constructed with straightedge and compass.

In this work Gauss used comprehensive approximation methods which he created for that purpose.

His friend Farkas Wolfgang Bolyai with whom Gauss had sworn " brotherhood and the banner of truth " as a student, had tried in vain for many years to prove the parallel postulate from Euclid's other axioms of geometry.

In 1831 Gauss developed a fruitful collaboration with the physics professor Wilhelm Weber, leading to new knowledge in magnetism ( including finding a representation for the unit of magnetism in terms of mass, length and time ) and the discovery of Kirchhoff's circuit laws in electricity.

In 1854, Gauss notably selected the topic for Bernhard Riemann's now famous Habilitationvortrag, Über die Hypothesen, welche der Geometrie zu Grunde liegen.

When his second wife died in 1831 after a long illness, one of his daughters, Therese, took over the household and cared for Gauss until the end of his life.

Therese kept house for Gauss until his death, after which she married.

They had an argument over a party Eugene held, which Gauss refused to pay for.

This is justified, if unsatisfactorily, by Gauss in his " Disquisitiones Arithmeticae ", where he states that all analysis ( i. e., the paths one travelled to reach the solution of a problem ) must be suppressed for sake of brevity.

* In Canadian junior high schools, an annual national mathematics competition ( Gauss Mathematics Competition ) administered by the Centre for Education in Mathematics and Computing is named in honour of Gauss,

However, the Theorema Egregium of Carl Friedrich Gauss showed that already for surfaces, the existence of a local isometry imposes strong compatibility conditions on their metrics: the Gaussian curvatures at the corresponding points must be the same.

The method is based on the individual work of Carl Friedrich Gauss ( 1777 – 1855 ) and Adrien-Marie Legendre ( 1752 – 1833 ) combined with modern algorithms for multiplication and square roots.

Gaussian elimination alone is sufficient for many applications, and requires fewer calculations than the Gauss – Jordan version.

Carl Friedrich Gauss in 1810 devised a notation for symmetric elimination that was adopted in the 19th century by professional hand computers to solve the normal equations of least-squares problems.

The algorithm that is taught in high school was named for Gauss only in the 1950s as a result of confusion over the history of the subject.

The IEC was instrumental in developing and distributing standards for units of measurement, particularly Gauss, Hertz, and Weber.

The Gauss algorithm for matrix inversion is probably the oldest solution but this approach does not efficiently use the symmetry of R and r. A faster algorithm is the Levinson recursion proposed by Norman Levinson in 1947, which recursively calculates the solution.

The terms for his scholarship stipulated that he was to visit Gauss in Göttingen and then continue to Paris.

Other standard iterative methods for matrix equation solutions can also be used, for example the Gauss – Seidel method, where updated values for each patch are used in the calculation as soon as they are computed, rather than all being updated synchronously at the end of each sweep.

" Four normal distribution | Gaussian distributions in statisticsThis unproved statement put a strain on his relationship with János Bolyai ( who thought that Gauss was " stealing " his idea ), but it is now generally taken at face value.

The second law of error is called the normal distribution or the Gauss law.

The concept of Gaussian processes is named after Carl Friedrich Gauss because it is based on the notion of the normal distribution which is often called the Gaussian distribution.

This eigenfunction gives the probability of the occurrence of a given integer in a continued fraction expansion, and is known as the Gauss – Kuzmin distribution.

follows by summing the Gauss – Kuzmin distribution.

In mathematics, the Gauss – Kuzmin distribution is a discrete probability distribution that arises as the limit probability distribution of the coefficients in the continued fraction expansion of a random variable uniformly distributed in ( 0, 1 ).

The distribution is named after Carl Friedrich Gauss, who derived it around 1800, and Rodion Kuzmin, who gave a bound on the rate of convergence in 1929.

Their nonuniversality can be observed by noticing that, if any of these are used to code the Gauss – Kuzmin distribution or the Zeta distribution with parameter s = 2, expected codeword length is infinite.

On the other hand, using the universal Elias gamma coding for the Gauss – Kuzmin distribution results in an expected codeword length ( about 3. 51 bits ) near entropy ( about 3. 43 bits ).

James Clerk Maxwell and Elihu Thomson ( through the British Association for the Advancement of Science-BAAS ) introduced the Centimetre gram second system of units ( cgs ) in 1874, in order to derive electric and magnetic metric units, following the recommendation of Carl Friedrich Gauss in 1832.

Gauss ' law states that " the total electric flux through any closed hypothetical surface of any shape drawn in an electric field is proportional to the total electric charge enclosed within the surface ".

This relation is known as Gauss ' law for electric field in its integral form and it is one of the four Maxwell's equations.

The electric field is zero outside of the depletion width ( seen in above figure ) and therefore Gauss ’ s law implies that the charge density in each region balance – as shown by the first equation in this sub-section.

According to Gauss ’ s law, a conductor at equilibrium carrying an applied current has no charge on its interior.

Smale's classification of immersions of spheres as the homotopy groups of Stiefel manifolds, and Hirsch's generalization of this to immersions of manifolds being classified as homotopy classes of maps of frame bundles are much further-reaching generalizations, and much more involved, but similar in principle – immersion requires the derivative to have rank k, which requires the partial derivatives in each direction to not vanish and to be linearly independent, and the resulting analog of the Gauss map is a map to the Stiefel manifold, or more generally between frame bundles.

This was a major discovery in an important field of mathematics ; construction problems had occupied mathematicians since the days of the Ancient Greeks, and the discovery ultimately led Gauss to choose mathematics instead of philology as a career.

Gauss ordered a magnetic observatory to be built in the garden of the observatory, and with Weber founded the " Magnetischer Verein " ( magnetic club in German ), which supported measurements of Earth's magnetic field in many regions of the world.

Its major part resembles the field of a bar magnet (" dipole field ") inclined by about 10 ° to the rotation axis of Earth, but more complex parts (" higher harmonics ") also exist, as first shown by Carl Friedrich Gauss.

* Carl Friedrich Gauss pioneers the field of summation with the formula summing 1: n as ( n ( n + 1 ))/ 2, at the age of 7.

* Carl Friedrich Gauss developed a fruitful collaboration with the physics professor Wilhelm Weber ; it led to new knowledge in the field of magnetism.

In 1833, Carl Friedrich Gauss, head of the Geomagnetic Observatory in Göttingen, published a paper on measurement of the Earth's magnetic field.

In contrast, one of the basic theorems in algebraic number theory asserts that the class group of the ring of integers of a number field is finite ; its cardinality is called the class number and it is an important and rather mysterious invariant, notwithstanding the hard work of many leading mathematicians from Gauss to the present day.

The average magnetic field strength in the Earth's outer core was measured to be 25 Gauss, 50 times stronger than the magnetic field at the surface.

The average magnetic field in the Earth's outer core was calculated to be 25 Gauss, 50 times stronger than the field at the surface.

The Earth's magnetic field strength was measured by Carl Friedrich Gauss in 1835 and has been repeatedly measured since then, showing a relative decay of about 10 % over the last 150 years.

In the case of a gravitational field g due to an attracting massive object, of density ρ, Gauss ' law for gravity in differential form can be used to obtain the corresponding Poisson equation for gravity.

* Milligauss ( mG ) and megagauss ( MG ), multiples of a unit of magnetic field Gauss ( unit ), equivalent to 1. 0 × 10 < sup >− 7 </ sup > tesla and 100 teslas, respectively.

The origins of class field theory lie in the quadratic reciprocity law proved by Gauss.

The Artin reciprocity law, which is a high level generalization of the Gauss quadratic reciprocity law, states that the product vanishes on the multiplicative group of the number field.

Also, the application of the theories of Carl Friedrich Gauss to magnetic observations showed that Earth's magnetic field had an internal, rather than external, origin.

* Carl Friedrich Gauss, at the age of seven, pioneers the field of summation with the formula summing 1: n as ( n ( n + 1 ))/ 2.

( Gauss sums are in a sense the finite field analogues of the gamma function.

Examples of complete exponential sums are Gauss sums and Kloosterman sums ; these are in some sense finite field or finite ring analogues of the gamma function and some sort of Bessel function, respectively, and have many ' structural ' properties.