Using matrix notation, the first fundamental form becomes

: Using knowledge of San beliefs, researchers have shown that the art played a fundamental part in the religious lives of its San painters.

# Using audio and later video, quotes from candidates are a fundamental part of the programme and interconnection with other media sources and impartial educational organisations will be sought.

Using newly development spectrophotometric equipment and modifications of the experimental procedure König and Dieterici published a more detailed paper in 1892, determining the “ fundamental sensations ” not only of subjects with normal color vision ( trichromats ) but also of dichromats and monochromats.

Using chemical and isotopic analyses, division scientists study samples of air, water, biological remains, rocks and meteorites in order to address a broad range of scientific issues, ranging from the particulate and chemical pollutants emitted by the collapse of the World Trade Center, to changes in earth's past climate, to the fundamental chemical processes involved in the differentiation and formation of the planet's mantle and core.

Using the above theorem it is easy to see that the original Borsuk Ulam statement is correct since if we take a map f: S < sup > n </ sup > → ℝ < sup > n </ sup > that does not equalize on any antipodes then we can construct a map g: S < sup > n </ sup > → S < sup > n-1 </ sup > by the formula

Using Wick theorem on the terms of the Dyson series, all the terms of the S-matrix for quantum electrodynamics can be computed through the technique of Feynman diagrams.

Using a uniqueness theorem and showing that a potential satisfies Laplace's equation ( second derivative of V should be zero i. e. in free space ) and the potential has the correct values at the boundaries, the potential is then uniquely defined.

Using the parallel axis theorem, the radius of gyration L of a rigid pendulum can be shown to be

Using the formula for Bayes ' theorem, we have:

Using the disintegration theorem we can define a restriction of Lebesgue measure to the-dimensional affine subspace of where the Gaussian distribution is supported, i. e..

Using Fibonacci numbers, he proved that when finding the greatest common divisor of integers a and b, the algorithm runs in no more than 5k steps, where k is the number of ( decimal ) digits of b. He also proved a special case of Fermat's last theorem.

Using Bayes ' theorem, we write

Using the above rule, we can complete the proof of Fermat's little theorem quite easily, as follows.

Using Noether's theorem, the types of Lagrangians that conserve X through a continuous symmetry may be determined, and their fitness judged by further criteria.

Using the Pythagorean theorem twice, b < sup > 2 </ sup > =

Using Dirichlet's theorem on primes in arithmetic progressions, the law of quadratic reciprocity, and the Chinese remainder theorem ( CRT ) it is easy to see that for any M > 0 there are primes p such that the numbers 1, 2, …, M are all residues modulo p.

Using Gauss's theorem and Poisson's equation gives

Using the close correspondence between divisors and holomorphic line bundles on a Riemann surface, the theorem can also be stated in a different, yet equivalent way: let L be a holomorphic line bundle on X.

Using topological cobordism and K-homology one may provide a full statement of an index theorem on quasiconformal manifolds ( see page 678 of ).

Using the compactness theorem, the resulting theory is easily seen to be consistent.

This theorem was never published by Desargues ( 1591 – 1661 ), but appeared in an appendix entitled Universal Method of M. Desargues for Using Perspective ( Maniére universelle de M. Desargues pour practiquer la perspective ) of a practical book on the use of perspective published in 1648 by his friend and pupil Abraham Bosse ( 1602 – 1676 ).

Using the theorem we can treat every abelian category as if it is the category of R-modules concerning theorems about existence of morphisms in a diagram and commutativity and exactness of diagrams.

Using Bloch's theorem, we only need to find a solution for a single period, make sure it is continuous and smooth, and to make sure the function u ( x ) is also continuous and smooth.

Using case 3 of the master theorem ( with ε = 1 and c = 2 / 3 ), we know that T ( n ) = Θ ( n ).

Using a theorem of Schwartz on compact operators in Frechet spaces, Cartan and Serre proved that compact complex manifolds have the property that their sheaf cohomology for any coherent sheaf consists of vector spaces of finite dimension.

Using Taylor's theorem,

Using the Riesz representation theorem, one can show that there exists a unique continuous linear operator

Using a branch of mathematics known as tuple calculus, he demonstrated that such a system could support all the operations of normal databases ( inserting, updating etc.

Using these two examples, we can make some useful observations that motivate the major ideas in lambda calculus.

Using the techniques of differential calculus, the velocity and acceleration as a function of time can be found:

Using differential calculus, we can determine the limit, or the value that Δy / Δx approaches as Δy and Δx get closer to zero ; it follows that this limit is the exact slope of the tangent.

Using the probability calculus of Bayes Theorem, Salmon concludes that it is very improbable that the universe was created by the type of intelligent being theists argue for.

Using geometry and calculus, some simple combinations of sources are easily solved ; others are not.

Using a method called " calculus of statement ," Lentz helps the team to mitigate the pressure harming the plant operators.

Using this in the sum, one can evaluate the remaining integral using the calculus of residues and arrive at the desired generating function.

Using the limit notation of calculus, the volume may be expressed as:

Using tensor calculus, proper time is more rigorously defined in general relativity as follows: Given a spacetime which is a pseudo-Riemannian manifold mapped with a coordinate system and equipped with a corresponding metric tensor, the proper time experienced in moving between two events along a timelike path P is given by the line integral

Using differential calculus we can obtain conditions on x under which this holds.

Using the parametrization of the hyperbola with hyperbolic angle, the theory of special relativity provided a calculus of relative motion with the one-parameter group indexed by rapidity.

* Using process calculus to model biological systems ( stochastic-calculus, BioAmbients, Beta Binders, BioPEPA, Brane calculus ).

Using calculus, Gershun could compute the irradiance falling on points () on a surface.

Using the functional calculus for larger classes of functions, we can for example define rigorously the " square root " of the ( negative ) Laplacian operator − Δ or the exponential

Using the bounded functional calculus, one can prove part of the Stone's theorem on one-parameter unitary groups:

Using typed lambda calculus to implement formal systems on a machine.

Using modern predicate calculus, we quickly discover that the statement is ambiguous.