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Gauss's method involved determining a conic section in space, given one focus ( the Sun ) and the conic's intersection with three given lines ( lines of sight from the Earth, which is itself moving on an ellipse, to the planet ) and given the time it takes the planet to traverse the arcs determined by these lines ( from which the lengths of the arcs can be calculated by Kepler's Second Law ).
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Gauss's and method
Gauss's presumed method was to realize that pairwise addition of terms from opposite ends of the list yielded identical intermediate sums: 1 + 100 = 101, 2 + 99 = 101, 3 + 98 = 101, and so on, for a total sum of 50 × 101 = 5050.
The equations are then discretized onto a grid using the finite difference method and solved subject to the constraints of Gauss's law and Linear elasticity.
An early demonstration of the strength of Gauss's method came when it was used to predict the future location of the newly discovered asteroid Ceres.
* The squared-error loss function is widely used in statistics, following Gauss's use of the squared-error loss function in justifying the method of least squares.
Bertrand translated into French Carl Friedrich Gauss's work on the theory of errors and the method of least squares.
In one case he reportedly gave a method equivalent to Gauss's pivotal condensation.
Gauss's and determining
It solves a special case of Gauss's class number problem of determining the number of imaginary quadratic fields that have a given fixed class number.
Gauss's and section
Gauss's law can be stated using either the electric field E or the electric displacement field D. This section shows some of the forms with E ; the form with D is below, as are other forms with E.
Gauss's and space
Coulomb's law is actually a special case of Gauss's Law, a more fundamental description of the relationship between the distribution of electric charge in space and the resulting electric field.
An important property of the exponential map is the following lemma of Gauss ( yet another Gauss's lemma ): given any tangent vector v in the domain of definition of exp < sub > p </ sub >, and another vector w based at the tip of v ( hence w is actually in the double-tangent space T < sub > v </ sub >( T < sub > p </ sub > M )) and orthogonal to v, remains orthogonal to v when pushed forward via the exponential map.
However, Gauss's Law says that the divergence of any possible electric force field is zero in free space.
In free space, the electric displacement field is equivalent to flux density, a concept that lends understanding to Gauss's law.
Gauss's and given
While Columb's law ( as given above ) is only true for stationary point charges, Gauss's law is true for all charges either in static or in motion.
Gauss's circle problem does not deal with the Gaussian integers per se, but instead asks for the number of lattice points inside a circle of a given radius centered at the origin.
If the electric field is known everywhere, Gauss's law makes it quite easy, in principle, to find the distribution of electric charge: The charge in any given region can be deduced by integrating the electric field to find the flux.
The suitability of Stirling's, Bessel's and Gauss's formulae depends on 1 ) the importance of the small accuracy gain given by average differences ; and 2 ) if greater accuracy is necessary, whether the interpolated point is closer to a data point or to a middle between two data points.
It follows from an application of Gauss's Lemma that if A is the norm of then the distance, induced by the metric, between two close enough points on the curve γ, say γ ( t < sub > 1 </ sub >) and γ ( t < sub > 2 </ sub >), is given by
Gauss's inequality gives an upper bound on the probability that a value lies more than any given distance from its mode.
Gauss's and one
Gauss's personal life was overshadowed by the early death of his first wife, Johanna Osthoff, in 1809, soon followed by the death of one child, Louis.
Should there be a nontrivial factor dividing all the coefficients of the polynomial, then one can divide by the greatest common divisor of the coefficients so as to obtain a primitive polynomial in the sense of Gauss's lemma ; this does not alter the set of rational roots and only strengthens the divisibility conditions.
There are two versions of the first message sent by Gauss and Weber: the more official one is based on a note in Gauss's own handwriting stating that " Wissen vor meinen – Sein vor scheinen " (" knowing before opining, being before seeming ") was the first message sent over the electromagnetic telegraph.
Gauss's law is one of Maxwell's equations governing electromagnetism.
Gauss's law for magnetism, which is one of the four Maxwell's equations, states that the total magnetic flux through a closed surface is equal to zero.
It is one of the four Maxwell's equations which form the basis of classical electrodynamics, the other three being Gauss's law for magnetism, Faraday's law of induction, and Ampère's law with Maxwell's correction.
Gauss's story was well known in Doyle's time, and Ramanujan's story unfolded at Cambridge from early 1913 to mid 1914 ; The Valley of Fear, which contains the comment about maths so abstruse that no one could criticise it, was published in September 1914.
They achieve that by sometimes using the average of two differences where Newton's or Gauss's would use just one difference.
No one should quit using Lagrange's or Gauss's because of it.
In electromagnetism one can derive the energy density of a field from Gauss's law, assuming the curl of the field is 0.
: The fact that was easy to prove and led to one of Gauss's proofs of quadratic reciprocity.
They actually only contain one of the original eight — equation " G " ( Gauss's Law ).
This follows directly from Gauss's law, by integrating over a small rectangular pillbox straddling one plate of the capacitor:
, one of Gauss's last students and a historian of mathematics, who was summarizing a remark made by Gauss about Eisenstein in a conversation many years earlier.
Gauss's and with
The survey of Hanover fueled Gauss's interest in differential geometry, a field of mathematics dealing with curves and surfaces.
Daniel Kehlmann's 2005 novel Die Vermessung der Welt, translated into English as Measuring the World ( 2006 ), explores Gauss's life and work through a lens of historical fiction, contrasting them with those of the German explorer Alexander von Humboldt.
Gauss's law for magnetism: magnetic field lines never begin nor end but form loops or extend to infinity as shown here with the magnetic field due to a ring of current.
" Abel said famously of Carl Friedrich Gauss's writing style, “ He is like the fox, who effaces his tracks in the sand with his tail .”
Gauss's law applies to, and can be used with any physical quantity that acts in accord to, the inverse-square relationship.
This is Gauss's celebrated Theorema Egregium, which he found while concerned with geographic surveys and mapmaking.
Gauss's law has a close mathematical similarity with a number of laws in other areas of physics, such as Gauss's law for magnetism and Gauss's law for gravity.
Gauss's law is something of an electrical analogue of Ampère's law, which deals with magnetism.
The law is valid in the magnetostatic approximation, and is consistent with both Ampère's circuital law and Gauss's law for magnetism.
Fourier transforming Gauss's law with this form for displacement field:
An early example of a divide-and-conquer algorithm with multiple subproblems is Gauss's 1805 description of what is now called the Cooley-Tukey fast Fourier transform ( FFT ) algorithm, although he did not analyze its operation count quantitatively and FFTs did not become widespread until they were rediscovered over a century later.
To derive Green's theorem, begin with the divergence theorem ( otherwise known as Gauss's theorem ):
The definition of electrostatic potential, combined with the differential form of Gauss's law ( above ), provides a relationship between the potential Φ and the charge density ρ:
If is a unique factorization domain with field of fractions, then by Gauss's lemma is irreducible in, whether or not it is primitive ( since constant factors are invertible in ); in this case a possible choice of prime ideal is the principal ideal generated by any irreducible element of.
which is in agreement with the continuity equation because of Gauss's law:
When it came to deriving the electromagnetic wave equation from displacement current in his 1865 paper A Dynamical Theory of the Electromagnetic Field, he got around the problem of the non-zero divergence associated with Gauss's law and dielectric displacement by eliminating the Gauss term and deriving the wave equation exclusively for the solenoidal magnetic field vector.
This can also be expressed in terms of vector field quantities by taking the divergence of Ampère's law with Maxwell's correction and combining with Gauss's law, yielding:
0.171 seconds.