Using their reciprocals, two Euler products for the Möbius function are,

Using the equivalence between Euler angles and rotation composition, it is possible to change to and from matrix convention.

Using the Euler product one finds that

Using the Euler formula for hinged ends, and substituting A · r < sup > 2 </ sup > for I, the following formula results.

Using this inequality to eliminate F from the Euler characteristic leads to the inequality E ≤ 3V − 3.

Using the geometric axis as the primary axis of a new body-fixed coordinate system, one arrives at the Euler equation of a gyroscope describing the apparent motion of the rotation axis about the geometric axis of the Earth.

* Programming: Principles and Practice Using C ++ by Bjarne Stroustrup – Addison-Wesley Professional ; 1 edition ( December 29, 2008 ); ISBN 0-321-54372-1

* R. E. Bryant, S. M. German, and M. N. Velev, Microprocessor Verification Using Efficient Decision Procedures for a Logic of Equality with Uninterpreted Functions, in Analytic Tableaux and Related Methods, pp. 1 – 13, 1999.

Using the extension of the code, the encoded string 0011001011 can be grouped into codewords as 0 – 011 – 0 – 01 – 011, and these in turn can be decoded to the sequence of source symbols acabc.

Using Unicode characters, the faces < big >&# x2680 ; &# x2681 ; &# x2682 ; &# x2683 ; &# x2684 ; &# x2685 ;</ big >, can be shown in text using the range U + 2680 – U + 2685 or using decimal.

* Using an Agile Software Process with Offshore Development – ThoughtWorks ' experiences with implementing XP in large distributed projects

Using the overall SADC figures as a comparative benchmark, an even smaller number came to Malawi – 219, 000 in 1998, 254, 000 in 1999 and 222, 526 in 2000.

* Using the biomimetic strategy, researchers in the Harvard University Wyss Institute demonstrated in a mouse model that the drug coated nanoparticles can dissolve blood clots by selectively binding to the narrowed regions in the blood vessels – just like the platelets do.

* 1879 – Using a filament of carbonized thread, Thomas Edison tests the first practical electric incandescent light bulb ( it lasted 13½ hours before burning out ).

Using 3700 or 3800 stadia ( approximately 420 – 430 miles or 5. 3 °- 5. 4 °) north of Marseilles for a base line obtains a latitude of 64. 8 ° or 64. 9 ° for Thule, well short of the Arctic Circle.

* ( VEW04 ) " Using a Decompiler for Real-World Source Recovery ", M. Van Emmerik and T. Waddington, the Working Conference on Reverse Engineering, Delft, Netherlands, 9 – 12 November 2004.

Using the same type of equipment as scuba diving, the diver breathes from compressed air cylinders, which float on a free floating raft at the surface, allowing the diver only 20 – 30 feet ( 6 – 9 m ) of depth to travel.

Using an Irish-produced magnetic system – on which animated characters were attached to a painted background, and then photographed through a 45 degree mirror – he persuaded Peter Firmin, who was then teaching at the Central School of Art, to create the background scenes.

Using a coronagraph on the Subaru telescope in Hawaii in 2005, astronomers were able to further constrain the size of a planet orbiting Vega to no more than 5 – 10 times the mass of Jupiter.

* October 22 – Using a filament of carbonized thread, Thomas Edison tests the first practical electric light bulb ( it lasts 13½ hours before burning out ).

Using a water clock, waterwheel and a series of gears, the Court Astronomer Zhang Heng ( 78 – 139 CE ) was able to mechanically rotate his metal-ringed armillary sphere.

* RFC 1027 – Using ARP to Implement Transparent Subnet Gateways

Using panel data estimates of export and import equations for 17 countries, Jakob B. Madsen ( 2002 ) estimated the effects of increasing tariff and non-tariff trade barriers on worldwide trade during the period 1929 – 1932.

* Laser treatment – Using a special strong beam of light of one color ( laser ) to heal a damaged area.

* Photocoagulation – Using a special strong beam of light ( laser ) to seal off bleeding blood vessels such as in the eye.

" Remote Physiological Monitoring Using a Microwave Interferometer ", Biomed Sci Instr 3: 291 – 307, 1967.

Using the method of Lagrange multipliers, we have, hence

Using Lagrange multipliers, this problem can be converted into an unconstrained optimization problem:

Using Lagrange multipliers, this gives

Using the Fibonacci sequence would keep the number of grains within any one field at minimum.

Using Zorn's lemma, it can be shown that every field has an algebraic closure, and that the algebraic closure of a field K is unique up to an isomorphism that fixes every member of K. Because of this essential uniqueness, we often speak of the algebraic closure of K, rather than an algebraic closure of K.

Using the principle of equivalence, Einstein concluded that the same thing holds in any gravitational field, that the rate of clocks R at different heights was altered according to the gravitational field g. When g is slowly varying, it gives the fractional rate of change of the ticking rate.

Using an electric field, molecules ( such as DNA ) can be made to move through a gel made of agar or polyacrylamide.

Using spectropolarimetry, a magnetic field has been detected on the surface of Vega by a team of astronomers at the Observatoire du Pic du Midi.

Using this convention that wave is defined as right-handed because when one points one's right thumb in the same direction of the wave ’ s propagation, the fingers of that hand curl in the same direction of the field ’ s temporal rotation.

Using this convention, left or right handedness is determined by pointing one ’ s left or right thumb toward the source, against the direction of propagation, and then matching the curling of one's fingers to the temporal rotation of the field.

Using a mathematical formulation, let ( x < sub > 1 </ sub >, x < sub > 2 </ sub >, ..., x < sub > n </ sub >) be the input sequence of n distinct values over the finite field F ; then the codebook C created from the tuplets of values obtained by evaluating every polynomial ( over F ) of degree less than k at each x < sub > i </ sub > is

Using the equations for lines and circles, one can show that the points at which they intersect lie in a quadratic extension of the smallest field F containing two points on the line, the center of the circle, and the radius of the circle.

Using Galois theory, certain problems in field theory can be reduced to group theory, which is in some sense simpler and better understood.

Using light microscopy treponemes are only visible using Dark field illumination.

where r is the full displacement vector from the wire element to the point at which the field is being computed and r ̂ is the unit vector of r. Using this the equation can be equivalently written

Using soil maps, farmers can pursue two strategies to adjust field inputs:

Using a DC pulse as the electrical current, spark plasma, spark impact pressure, joule heating, and an electrical field diffusion effect would be created.

Using quantum mechanics it can be shown that for an electron in an external electric field E, the acceleration a < sub > ℓ </ sub > along coordinate direction ℓ is:

Using existing lenses meant that the field of view was reduced by around 1. 6 ×, but had the advantage of a larger lens selection.

Using a known electric field, Millikan and Fletcher could determine even the charge on oil droplets in mechanical equilibrium.

Using Ampere's law, one can determine the magnetic field associated with a given current or current associated with a given magnetic field, providing there is no time changing electric field present.

Using light cone gauge, yields light-cone string field theories whereas using BRST quantization, one finds covariant string field theories.